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On a Finite Element Approach to Modeling of Piezoelectr ic Element Dr iven Compliant Mechanisms
A Thesis
Submitted to the College of Graduate Studies and Research
in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
in the
Department of Mechanical Engineering
University of Saskatchewan
Saskatoon, Saskatchewan
Canada
By
RANIER CLEMENT TJIPTOPRODJO
Copyright Ranier Clement Tjiptoprodjo, April 2005. All rights reserved.
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PERMISSION TO USE
In presenting this thesis in partial fulfilment of the requirements for a Master of Science
degree from the University of Saskatchewan, the author agrees that the Libraries of this
University may make it freely available for inspection. The author further agrees that
permission for copying of this thesis in any manner, in whole or in part, for scholarly
purposes may be granted by the professor or professors who supervised the thesis work
or, in their absence, by the Head of the Department or the Dean of the College in which
the thesis work was done. It is understood that any copying or publication or use of this
thesis or parts thereof for financial gain shall not be allowed without the author’s
written permission. It is also understood that due recognition shall be given to the
author and to the University of Saskatchewan in any scholarly use which may be made
of any material in this thesis.
Requests for permission to copy or to make other use of material in this thesis in whole
or part should be addressed to:
Head of the Department of Mechanical Engineering
University of Saskatchewan
Saskatoon, Saskatchewan S7N 5A9 CANADA
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ABSTRACT
Micro-motion devices may share a common architecture such that they have a main
body of compliant material and some direct actuation elements (e.g., piezoelectric
element). The shape of such a compliant material is designed with notches and holes on
it, and in this way one portion of the material deforms significantly with respect to
other portions of the material – a motion in the conventional sense of the rigid body
mechanism. The devices of this kind are called compliant mechanisms. Computer tools
for the kinematical and dynamic motion analysis of the compliant mechanism are not
well-developed.
In this thesis a study is presented towards a finite element approach to the motion
analysis of compliant mechanisms. This approach makes it possible to compute the
kinematical motion of the compliant mechanism within which the piezoelectric
actuation element is embedded, as opposed to those existing approaches where the
piezoelectric actuation element is either ignored or overly simplified. Further, the
developed approach allows computing the global stiffness and the natural frequency of
the compliant mechanism.
This thesis also presents a prototype compliant mechanism and a test bed for measuring
various behaviors of the prototype mechanism. It is shown that the developed approach
can improve the prediction of motions of the compliant mechanism with respect to the
existing approaches based on a comparison of the measured result (on the prototype)
and the simulated result. The approach to computation of the global stiffness and the
natural frequency of the compliant mechanism is validated by comparing it with other
known approaches for some simple mechanisms.
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ACKNOWLEDGEMENTS
I need to acknowledge Dr. (Chris) W.J. Zhang that has been a superb and
dedicated research supervisor. I like to acknowledge his moral and financial support.
His vision and broad knowledge play an important role in this research work. I also like
to thank him for pushing me to the stage that I thought I never could accomplish. I also
like to thank my research co-supervisor, Dr. M. M. Gupta for his continuous moral
support. Allow me to express my sincere gratitude to Dr. A. T. Dolovich, my academic
and teaching mentor for his tremendous and sincere support during my study here. It is
my honor to have had opportunity to work with these great minds and their great
characters. Their supports, commitments and contributions have been an inspiration.
I need also to express my gratitude to the Department of Mechanical
Engineering through student assistantship experience, in which I have an opportunity to
gain some financial support. I like to thank these following people that have shared
their teaching experience: Dr. I. Oguocha, Dr. C. M. Sargent, Dr. P. B. Hertz and Dr. G.
J. Schoenau. Allow me to also acknowledge Dr. W. Szyszkowski and Dr. L. G. Watson
for sharing their expertise in finite element method and using ANSYS. Also, I
appreciate the moral support from Dr. R. T. Burton during my very first years of study.
I also like to thank Dr. D.X.B. Chen for his support.
Many thanks also go out to my colleagues in the AEDL. P. Ouyang that has
contributed significantly his expertise to help me completing the experiment results
presented in this thesis. Also I like to express my gratitude to K. D. Backstorm, B.
Zettl, and D. Handley for their supports that play important role in this thesis.
I also appreciate the supports from these people: A. Wettig, S. Haberman, D.
Bitner, D. Braun, C. Tarasoff, D. Vessey, A. McIntosh and C. Jansen. I like also to
acknowledge H. Sato from TOKIN Inc, J. Mueller from KAMAN, and H. Saadetian
and P. Budgell from ANSYS for their patience in answering all of my questions.
I like to acknowledge Heli Eunike for her continuous support and understanding
to deal with me during my tough times prior to completing this thesis. The last but
definitely not the least, I like to thank my family for their love and support, in particular
Herlina (mom), Lydiana (auntie), Kevin and Natalia (younger brother and sister).
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DEDICATION
To my parents:
Rudy Prabowo Tjiptoprodjo (Liem Kie Boan)† and Herlina Alimin (Lie Lian Hoa)
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TABLE OF CONTENTS
PERMISSION TO USE..............................................................................................i
ABSTRACT…………………………………………………………………………ii
ACKNOWLEDGMENTS………………………...……………………………….iii
TABLE OF CONTENTS…………………………………...……………………...iv
L IST OF FIGURES…………………...…………………………………………..vii
L IST OF TABLES…………………………………………………………………xi
CHAPTER 1 INTRODUCTION…………………………………………………..1
1.1 Research Background and Motivation …………………………………………..1
1.2 A Brief Review of the Related Studies ………………………………………….3
1.3 Research Objectives ……………………………………………………………..5
1.4 General Research Method ……………………………………………………….5
1.5 Organization of the Thesis ………………………………………………………6
CHAPTER 2 L ITERATURE REVIEW ………………………………………….7
2.1 Introduction ……………………………………………………………………...7
2.2 Piezoelectric Material and its applications ……………………………………...8
2.2.1 Piezoelectric materials [Setter, 2002] ……………………………………..8
2.2.2 Properties of PZT actuator [Tokin, 1996] ………………………………..11
2.2.3 PZT Actuator manufacturing and operation ……………………………..19
2.2.4 Modeling and analysis of PZT devices …………………………………..22
2.3 Compliant Mechanism …………………………………………………………24
2.4 Finite Element Analysis by use of ANSYS ………………………………….....28
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2.5 The Natural Frequency of the Compliant Mechanism …………………………31
2.6 The Stiffness of the Compliant Mechanism………………….………….……...34
CHAPTER 3 FINITE ELEMENT ANALYSIS OF DISPLACEMENT OF
RRR MECHANISM ………………………………………………37
3.1 Introduction ……………………………………………………………………..37
3.2 Fundamental Information ……………………………………………………….38
3.2.1 Multidisciplinary element type [ANSYS, 2004] ………………………….38
3.2.2 Piezoelectric material data ………………………………………………...42
3.3 Kinematic Analysis of the RRR Mechanism …………………….………….…..45
3.4 Modeling of the PZT Actuator for the RRR Mechanism ……………………..…51
3.5 Finite Element Modeling of the PZT-RRR Mechanism…………………….…...57
3.6 Illustrations………………………………………………………………….……64
3.7 Summary and Discussions…………………….………………………….………67
CHAPTER 4 NATURAL FREQUENCY AND STIFFNESS ……………………69
4.1 Introduction ………………………………………………………………………69
4.2 Natural Frequency of Compliant Mechanisms …………………………………..69
4.2.1 Basic concepts……………………………………….…………….……......69
4.2.2 Procedure……………………………………………………………………72
4.2.3 Validation…………………………………………………………….…......75
4.2.4 Results………………………………………………………………....……79
4.3 System Stiffness …………………………………………………………….…. 82
4.3.1 Basic concepts……………………………………………...………….........82
4.3.2 Procedure………………………………………………….………………..82
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4.3.3 Validation…………………………………………………………………..85
4.3.4 Results……………………………………………………………….……..88
4.4 Conclusion …………………………………………………………………….....91
CHAPTER 5 EXPERIMENTAL VALIDATION …………………………….....92
5.1 Introduction ……………………………………………………………………...92
5.2 Measurement Test-bed Set-up …………………………………………………..92
5.2.1 Measurement at the end-effector…………………………………………. 92
5.2.2 Measurement at the actuator level ………………………………………. 101
5.3 Results and Discussions ………………………………………………………. 101
5.4 Summary and Conclusion …………………………………………………….. 107
CHAPTER 6 CONCLUSIONS AND FUTURE WORK ……………………… 108
6.1 Overview ………………………………………………………………………. 108
6.2 Contributions of the Thesis ……………………………………………………. 111
6.3 Future Work …………………………………………………………………… 111
REFERENCES…………………………………………………………………… 113
APPENDIX A………………………………………………...…………………… 120
APPENDIX B………………………………………………...…………………….129
APPENDIX C………………………………………………...…………………… 132
APPENDIX D………………………………………………...…………………… 142
APPENDIX E………………………………………………...…………………….146
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LIST OF FIGURES
Figure Page
1-1. Schematic diagram of a RRR mechanism………..…………………………..2
1-2. Finite element model of the compliant mechanism [Zou, 2000]………….…..4
2-1. Unpolarized vs. polarized piezoelectric material……………………...………7
2-2. AE0505D16 [Tokin, 1996]……………………………………………….…..11
2-3. Direction and orientation axis of piezoelectric material……………….……..11
2-4. Impedance-frequency characteristic
of the piezoelectric actuator [Tokin, 1996]…………………………….……..14
2-5. The holes pattern for manufacturing sheets……………………………..........20
2-6. Stacked layers of piezoelectric actuator AE0505D16 [Tokin, 1996]……..…..21
2-7. Manufactured compliant piece [Zou, 2000]…………………………....….….25
2-8. Revolute joint of rigid body versus compliant body [Zou, 2000]……….........25
2-9. Finger tip sensor…………………………....…………………………….…...27
2-10. RRR mechanism……………………………………....………………….…...29
2-11. General procedures to perform FEA by use of ANSYS …............................. 31
3-1. Geometry of SOLID 5 [ANSYS, 2004]…………………………...……….…38
3-2. Disciplines in SOLID 5 [ANSYS, 2004]…………………………...…….…..39
3-3. Geometry of PLANE 13 [ANSYS, 2004]……………………..….……..…...40
3-4. Disciplines in PLANE 13 …………………………...……..……….…...……41
3-5. Degrees of freedoms ……………………………………….…………..….….42
3-6. Zou’s Finite Element Model of the RRR mechanism…………………...…....46
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3-7. Finite element model of the RRR mechanism in this thesis………………....47
3-8. The motion nature of the RRR mechanism……………………………….....49
3-9. The application of electrical input on the PZT actuators……………….…...50
3-10. Geometric boundaries of the PZT actuator…………………………….……52
3-11. Axis of piezoelectric material………………………………………….……55
3-12. Different types of meshing density on PZT………………………………...56
3-13. COMBIN 14 [ANSYS, 2004]………………………………………………58
3-14. The modeled PZT, plate, and compliant piece……………………………...61
3-15. Modeling the boundary conditions and the bolts…………………………...62
3-16. Modeling the end-effector platform………………………………………...63
3.17. The deformation of the RRR mechanism
by activating the single PZT actuator ………………………………........…65
3-18. The deformation of the RRR mechanism by activating two PZT actuators...66
3-19. The deformation of the RRR mechanism by activating all PZT actuators….67
4-1. The procedure to compute the system frequency……………………..……..74
4-2. A four-bar mechanism………………………………………………….……75
4-3. Results comparison for the first mode…………………………….……...….76
4-4. Results comparison for the second mode……….……………….…………..77 4-5. Results comparison for the third mode………………………….…….......…78 4-6. The natural frequencies of the PZT 1 actuation….…….….…………………80 4-7. The natural frequencies of the PZT 2 actuation ….….…................................80 4-8. The natural frequencies of the PZT 3 actuation ….….…................................81 4-9. The natural frequencies of the PZT 1 and 2 actuation ……….……..…........81 4-10. The natural frequencies of the PZT 1 and 3 actuation …..…...…………......82
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4-11. The natural frequencies of the PZT 2 and 3 actuation …..…...…………......82 4-12. The natural frequencies of the PZT 1, 2 and 3 actuation …..……………......82 4-13. The PRBM of the RRR mechanism [Zou, 2000]……… …..……………......86 4-14.A two-legged planar manipulator……………………… …..…………….......86 4-15. The stiffness of the PZT 1 actuation………………………………………….90 4-16. The stiffness of the PZT 2 actuation………………………………………….91 4-17. The stiffness of the PZT 3 actuation………………………………………….91 4-18. The stiffness of the PZT 1 and 2 actuation…………………..….……………92 4-19. The stiffness of the PZT 2 and 3 actuation…………………..….……………92 4-20. The stiffness of the PZT 1 and 3 actuation…………………..….……………93 4-21. The stiffness of the PZT 1, 2, and 3 actuation…………...…..….…….…..….93 5-1. SMU 9000-15N-001 [Kaman, 2000]………………………………………....93 5-2. Eddy current behavior [Kaman, 2000]……………………….…….………....94 5-3. Required distance between sensor and target [Kaman, 2000].….……....….....94 5-4. Sensor mounting requirement [Kaman, 2000].……………………………......95 5-5. Parallelism requirement [Kaman, 2000]………………………………….……95 5-6. Target requirements [Kaman, 2000]………..…………………………….........95 5-7. Requirement for sensor to sensor proximity [Kaman, 2000] ………….….......96 5-8. Calibration of SMU 9000-15N-001 for the RRR mechanism application……97 5-9. The RRR mechanism set-up in experiment………………..……....…...…......98 5-10. Adjustable workbench……………………..…………………….……….......99 5-11. The reference in simulation versus the reference in measurement…………..100 5-12. A schematic diagram of the measurement system [Handley, 2002].…..…....101
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5-13. Rotating the positions of A, B and C…………………….……...….….……...102
5-14. Check the data repeatability of
the end-effector deformations in experiment….…….....…………….……......103
5-15. Comparison of the end-effector deformation results…….…….………..... ….105
5-16. Comparison between measurement and simulation at the actuator level...........106
A-1. The assembly of the RRR mechanism………………………………….….......120
A-2. A main body…………………………………………………….…………..….122
A-3. The attached strain gage on the PZT actuator………………………….………124
A-4. The end-effector platform……………………………………………………...125
A-5. Geometric boundaries of a bolt…………………………………………….......126
A-6. Location of the thin metal plate within the RRR………………………………126
A-7 Compliant piece under prestress state………………………………………….127
B-1. The slots of the PZT actuators………………………………....………………129
B-2. The pre-deformed slots of the PZT actuators………………………………….130
C-1. Required distance between sensor and target [Kaman, 2000]….…...………....132
C-2. Distance between sensor and target in the workbench………………………...133
C-3. Sensor mounting requirement…………………………………...…………......133
C-4 Geometric boundaries of dovetail and sensor……..…………………………...134
C-5 The height of the sensor…………………...……...…………………………....135
C-6. Parallelism requirement [Kaman, 2000]……………..…………………..........135
C-7. Solution to compensate the parallelism requirement……..…………………...136
C-8. Target requirements [Kaman, 2000]…………..………………………………136
C-9. Geometric boundaries of a target……………..………………………….........137
C-10. Requirement for sensor to sensor proximity…………………………….….....138
C-11. The distance between two sensors…...………………………………………..138
C-12. An assembled workbench…………………………………………………..…139
C-13. Components of a workbench……………………………………………….....140
D-1. Find the accurate distances between sensors and targets……………………..142
D-2. The X-Y-Z stage (M-461, Newport Company)………………………………143
D-3. The measuring process of the end-effector displacements………………...…144
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LIST OF TABLES
Table Page
4-1. Table of structural elements as P (10,10) cm……………………………..…42
5-1. The configurations of the RRR mechanism (all PZT are activated)…….…..44
D-1. The smallest errors for the three sensors……………………………….….144
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LIST OF ABBREVIATIONS
AEDL Advanced Engineering Design Laboratory
DC Direct current
FEM/FEA Finite element model/finite element analysis
PZT Piezoelectric actuator that has material consists of ions Pb, Zr and Ti.
PRBM Pseudo rigid body model
RRR Revolute revolute revolute (a given name to the compliant mechanism
based on its PRBM)
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CHAPTER 1
INTRODUCTION
1.1 Research Background and Motivation
In applications such as chip assembly in the semiconductor industry, cell
manipulation in biotechnology, and surgery automation in medicine, there is a need
for a device to perform controlled small motion (less than 100 µm) with high
positioning accuracy (in the submicron range) and complex trajectories. This range of
motion is known as micro-motion [Hara and Sugimoto, 1989]. The need of such a
kind of device is also found in many intelligent devices which have the capability of
sensing and making decisions in response to external disturbances.
The devices of this kind share a common architecture as follows. The devices have a
compliant main body, the shape of which is designed with notches and holes on it.
One portion of the material deforms significantly with respect to other portions of the
material and illustrates or results in a sort of motion in the conventional sense of the
rigid body mechanism. It was reported that systems built based on the compliant
structure concept make it possible to achieve 0.01 µm positioning accuracy [Hara and
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Sugimoto, 1989; Her and Chang, 1994]. The devices of this kind are called compliant
mechanisms. Driving components in the compliant mechanism are usually developed
by means of the piezoelectric technology (PZT for short), because of its advantages
of fast response, and smooth and high-resolution displacement characteristics [Lee
and Arjunan, 1989]. The PZT actuator used in this thesis is capable in achieving a
displacement of 15 µm, while its resolution is sub-nanometer.
The compliant structure incorporating the actuator is called the compliant
mechanism. A compliant mechanism can be configured as a closed-loop layout. The
closed-loop configuration can provide better stiffness and positioning accuracy.
Figure 1.1 shows one example of a compliant mechanism.
Figure 1.1 Schematic diagram of a RRR mechanism.
PZT 3
75.71 mm
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This mechanism consists of a compliant main body and a member of rigid material
which is geometrically an equilateral triangle. The mechanism is driven by three PZT
actuators (PZT 1, PZT 2 and PZT 3), while its end-effector motion is located at the
center point O of the rigid member (Fig. 1.1). This mechanism is typically used to
produce planar micro-motions with two translations (x and y) and one rotation (θ)
and has been found in applications in the semiconductor industry [Ryu et al., 1997].
It is noted that in industrial applications, the terms micro-positioning stage and
single-axis stage are used. They represent a kind of micro-motion system, and thus
they are used interchangeably with the term micro-motion system in this thesis.
It is important to develop a model for the micro-motion device in order to simulate or
predict behavior and performance of the device. The behaviors important to functions
are the motion, stiffness, and natural frequency. For the micro-motion system, a large
motion range is pursued; yet the large motion range may compromise the system
stiffness. The information of the natural frequency is useful to determine the speed
range of the PZT actuator such that the resonant situation can be avoided.
In this thesis, the compliant mechanism shown in Fig. 1.1 is studied
comprehensively, and this compliant mechanism is thereafter called the RRR
mechanism.
1.2 A Br ief Review of the Related Studies
There have been several studies at the Advanced Engineering Design Laboratory at
the Department of Mechanical Engineering at the University of Saskatchewan. Zou
[2000] pioneered a study on the mechanism as shown in Figure 1.1. The work by Zou
[2000] has not modeled the physical behaviour of piezoelectric actuators. In addition,
the finite element model using the triangular type of element appears to contain some
bad-shaped elements: refer to Figure 1.2. A popular approach, called pseudo rigid
body (PRB) method, for compliant mechanisms, was also applied to kinematic and
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dynamic analysis from Ref. [Zou, 2000]. It has been shown that the equation for the
dynamic motion analysis is extremely complex, containing 600 lines of strings with
the Maple V software [Maple, 1997].
Zettl [2003] developed a more effective 2D finite element model for the same
compliant mechanism. The author led to a drastic reduction of the computational time
for the motion analysis of the compliant mechanism yet without sacrificing prediction
accuracy. In the study performed by Zettl [2003], consideration of the physical
property of the PZT actuators is not systematic in the sense that the properties of the
piezoelectric material were not fully explored. Only because conventional types of
elements, e.g., spring, truss, or beam, was applied in his work.
The modeling method developed by Zou [2000] for this compliant mechanism has
been verified by experimental measurement. However, the previous experimental set
up and the measurement technique for this compliant mechanism [Zou, 2000] were
not very reliable. Furthermore, neither of these two studies has provided a tool for the
Figure 1.2 Finite element model of the compliant mechanism [Zou, 2000].
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simulation of the system stiffness and the natural frequency. Those studies did not
consider the prestress in the PZT actuator either.
1.3 Research Objectives
The primary goal of the study presented in this thesis was to improve the above
methods and develop a method for the simulation of the system stiffness and the
natural frequency. The secondary goal was to develop a more reliable test bed for the
validation of the model for motion analysis. The following research objectives were
defined.
Objective 1: To develop a more accurate finite element model of the compliant
mechanism (see Fig. 1.1) for motion analysis with special attention to capturing the
physical behaviour of the piezoelectric actuators with the compliant mechanism.
Objective 2: To develop a more reliable test bed for the compliant mechanism (see
Figure 1.1) with the objective to provide a test environment for the validation of the
model for motion analysis.
Objective 3: To develop methods based on finite element analysis for predicting the
system stiffness and natural frequency properties.
1.4 General Research Method
The basic idea underlying this research is to apply a general-purpose finite element
tool, ANSYS, in which several special types of elements are provided for the so-
called multidisciplinary field or effect including the coupling of the mechanical
displacement and electrical current (PZT actuator or sensor). The use of the finite
element analysis for the compliant mechanism is a natural choice because the
compliant material is by itself better to be viewed as an object with material
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continuity. In other words, the compliant material is not lumped inherently. This
means that the PRB method is inherently not suitable to the motion analysis of the
compliant mechanism.
1.5 Organization of Thesis
This thesis consists of six chapters. Some general idea of discussions on each chapter
will be concisely described as follows.
Chapter 2 discusses background for this research and provides a literature review.
The literature review is focused on the PZT compliant mechanism and the
methodology used for its analysis. The discussion in Chapter 2 further confirms the
need of the research described in this thesis.
Chapter 3 presents a finite element model for the motion analysis of the PZT-RRR
mechanism. The model is expected to overcome the shortcomings in the study by
Zou [2000] and Zettl [2003]. An illustration is given to see how the simulation of
motion can be generated with this model.
Chapter 4 presents finite element methods for the calculation of the system stiffness
and the natural frequency.
Chapter 5 presents the development of a test bed for the verification of the finite
element model for motion analysis developed in Chapter 3. A comparison is made
between the three theoretical methods, namely the one developed in this thesis, the
one developed by Zou [2000], and the one developed by Zettl [2003]. The
experimental measurement will also be described.
Chapter 6 concludes this thesis with discussion of the results, contributions, and
future work.
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CHAPTER 2
BACKGROUND AND
LITERATURE REVIEW
2.1 Introduction
This chapter provides both a literature review and the background necessary to
facilitate the understanding this thesis, in particular its proposed research objectives
and scope discussed in Chapter 1. Section 2.2 introduces the piezoelectric material
and its applications, as well as the use of the piezoelectric material as actuators.
Section 2.3 describes a compliant mechanism in more detail and explains the reasons
behind using this specific type of compliant mechanism for micro-manipulation.
Section 2.4 introduces how a particular finite element analysis software package
ANSYS addresses the problem which combines different disciplinary domains, in
particular the modeling of PZT actuators embedded in a structure. Section 2.5
discusses the concepts of system stiffness and natural frequency and the current
method of calculating them.
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2.2 Piezoelectr ic Mater ial and its Applications
2.2.1 Piezoelectr ic Mater ials [Setter , 2002]
Piezoelectric materials have found applications in a wide range of fields, such as
medical instrumentations, industrial process control, semiconductor manufacturing
system, household electrical appliances, and environmental monitoring
communications. Commercial equipment systems that use piezoelectric materials are
found in pumps, sewing machines, pressure sensors, optical instruments, heads for
dot and ink jet printers, and linear motors for camera auto focusing. The range of
applications continues to grow.
Applications of piezoelectric materials generally fit into four categories: sensors,
generators, actuators, and transducers. In the generator category, piezoelectric
materials can generate voltages that are sufficient or larger to spark across an
electrode gap, and thus can be used as ignitors in fuel lighters, gas stoves, and
welding equipment. Alternatively, the electrical energy generated by a piezoelectric
element can be stored. Such generators are excellent solid state batteries for
electronic circuits. In the sensor category, piezoelectric materials convert a physical
parameter, such as acceleration, pressure, and vibrations, into an electrical signal. In
the actuator category, the piezoelectric materials convert an electrical signal into an
accurately controlled physical displacement, to finely adjust precision machining
tools, lenses, or mirrors. In the transducer category, piezoelectric transducer can both
generate an ultrasound signal from electrical energy and convert an incoming sound
into an electrical signal. Piezoelectric transducer devices are designed for measuring
distances, flow rates, and fluid levels. The piezoelectric transducers are used to
generate ultrasonic vibrations for cleaning, drilling, welding, milling ceramics, and
also for medical diagnostics.
In the year of 1880, Jacques and Pierre Curie discovered an unusual characteristic of
certain crystalline minerals: when subjected to the mechanical force, they became
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electrically polarized. Subsequently, the inverse of this relationship was confirmed: if
one of these voltage-generating crystals was exposed to an electric field, it
lengthened or shortened according to the polarity of the field, and in proportion to the
strength of the field. These behaviors were labeled the piezoelectric effect and the
inverse piezoelectric effect, respectively. A piezoelectric material possesses a
crystalline structure of lead zirconate titanate PbZrO3-PbTiO3 (abbreviated to PZT),
which is the primary component of the piezoelectric material. The crystalline
structure of the PZT controls the behavior of the piezoelectric material. The
behaviors of the PZT actuator with respect to the crystalline structure can be
classified into two conditions, unpolarized and polarized piezoelectric material (as
illustrated in Fig. 2.1). In the unpolarized piezoelectric material condition (see Fig.
2.1a.), Ti and Zr ions are centered on the lattice (the arrangement of ions or
molecules within the crystal). At this time, the piezoelectric material is electrically
balanced and neither electrical polarization nor mechanical deformation arises in the
material. Such a condition occurs when one does not apply electrical voltages on the
piezoelectric material, or when one applies electrical voltages on the piezoelectric
material in the temperature that exceeds the Curie temperature. The Curie
temperature is a temperature that limits the piezoelectric material such that when the
piezoelectric material is operated above this temperature, it will cease to work. In the
polarized piezoelectric material condition, Ti and Zr ions are no longer centered on
the lattice, due to the applied electrical field that causes the axis of the crystal to
become longer in the direction parallel to the direction of the applied electric field.
The specific behavior of the crystal also influences the neighboring crystals such that
the entire domain behaves similarly (see Fig. 2.1b.). Such behavior occurs when one
applies the electrical voltages to the piezoelectric material without exceeding the
Curie temperature.
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Figure 2.1 Unpolarized vs. polarized piezoelectric material.
Actuators made of the piezoelectric material are used in the RRR mechanism. The
manufacturer of the actuator is Tokin America Inc. This actuator consists of multiple
layers of piezoelectric sheets. The model name of the actuator is AE0505D16 (see
Fig. 2.2). In the following, the properties of the PZT actuator, taking the
AE0505D16 as an example, are discussed. The general knowledge is largely drawn
from Ref. [Setter, 2002]; while the specific knowledge related to the actuator
(AE0505D16) is based on its manufacturer [Tokin, 1996].
Figure 2.2 AE0505D16 [Tokin, 1996].
b
a
l
Longitudinal or actuating direction
-
+
(+)
(-)
+
-
(a) (b)
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2.2.2 Properties of PZT actuator [Tokin, 1996]
By their nature, piezoelectric materials are anisotropic. Fig. 2.3 denotes the different
direction and orientation axis of the piezoelectric material. In order to facilitate the
understanding of the material properties of the piezoelectric actuators, those axes are
explained. Axes 1, 2, and 3 are consecutively analogous to X, Y, Z of the classical
right hand orthogonal axial set, while axes 4, 5, and 6 identify the rotations’ axes.
The direction of axis 3 is the direction of polarization. Polarization is the process that
occurs when an electric field is applied between two electrodes. For actuator
applications, the largest deformation is along the polarization axis (i.e., axis 3).
`
Figure 2.3 Direction and orientation axis of piezoelectric material.
The material properties of the piezoelectric actuator are listed below:
1. Relative dielectric constant,
2. Frequency constant,
3. Electromechanical coupling constant,
4. Elastic constant,
5. Piezoelectric constant,
6. Poisson’s ratio,
1
2
3
4 5
6 Polarization
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12
7. Temperature coefficient,
8. Aging rate,
9. Mechanical quality factor,
10. Curie temperature, and
11. Density
(1) Relative dielectric constant
0
33)(
εε ST
= 5440
0
11)(
εε ST
= 5000
where ε 0 is the dielectric permittivity of a vacuum (= 8.85 x 10-12 Farads/meter).
Relative dielectric constant is the ratio of the dielectric permittivity of the material (in
this case, εT33 and εT
11) to the dielectric permittivity of a vacuum (ε0). The
superscripts denote the boundary condition on material as the process of
determination of the relative dielectric constant values; specifically the superscript T
(in this case) describes the condition of constant stress (not clamped). Note that the
superscript S refers to the condition where constant strains are measured.
As for the subscripts of the relative dielectric constant, the first subscript indicates the
direction of dielectric displacement and the second subscript indicates the direction of
electrical field. A formula to obtain the relative dielectric constant is given as follows
[Tokin, 1996]:
0εε ij
T
=S
tC
0ε (2.1)
(AE0505D16)
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13
where ijTε : valid for either 11
Tε or 33Tε ,
t : distance between electrodes (m),
S : electrode area (m2),
and C : static capacitance (Farads).
(2) Frequency constant
N3 = 1370 Hz-m (AE0505D16)
When an electrical voltage is applied to the piezoelectric actuator, the resulting
frequency should be well below its resonance frequency. Otherwise, the actuator will
vibrate in an uncontrollable manner. The directions of polarizations and vibrations
are along the longitudinal axis in the core of the PZT actuator. A formula to obtain
the frequency constant is given as follows [Tokin, 1996]:
lf r ×=Ν3 (2.2)
where 3Ν : frequency constant,
rf : resonance frequency = 68500 Hz,
and l : the length of AE0505D16 = 20 mm.
(3) Electromechanical coupling constant
K longitudinal = 0.68 (AE0505D16)
A formula to obtain the electromechanical coupling constant for the longitudinal
vibration is [Tokin, 1996]:
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14
K longitudinal= ).2
cot().2
(a
r
a
r
f
f
f
f ππ (2.3)
where fr : resonant frequency (68500 Hz),
and fa : anti-resonant frequency (79400 Hz).
The coefficient of electromechanical coupling is defined as the mechanical energy
accumulated in a material, which is related to the total electrical input. This
coefficient can be calculated by measuring the resonant and the anti-resonant
frequencies. To measure those frequencies, an impedance analyzer is commonly used
to depict the impedance-frequency characteristic of the piezoelectric actuator (see
Fig. 2.4).
Figure 2.4 Impedance-frequency characteristic of the piezoelectric actuator
[Tokin, 1996].
By its nature, the resonance frequency occurs when the system has very small
resistance, while the anti-resonance frequency occurs when the system has very large
resistance. In Fig. 2.4, the frequency that minimizes the impedance is chosen as
resonant frequency (fr) and the frequency that maximizes the impedance is chosen as
anti-resonance frequency (fa).
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15
(4) Elastic constant
SE(D)11=14.8 x 10-12 m2/N
SE(D)33=18.1 x 10-12 m2/N
Elastic constant (S) defines the strain due to an applied stress (compliance). The
superscripts denote the imposed conditions on material. The superscript E describes
the boundary condition of the constant electrical field (the electrodes connected
together or short circuit), while the superscript D indicates the boundary condition of
the constant dielectric displacement (no current flows or open circuit). As for two
digits in subscripts, they represent the directions of stress and strain. The first
subscript indicates the direction of strain, and the second subscript indicates the
direction of stress.
(5) Piezoelectric constant
d31= -287 x 10-12 m/V
d33 = 635 x 10-12 m/V
d15 = 930 x 10-12 m/V
g31 = -6 x 10-3 Vm/N
g33 = 13.2 x 10-3 Vm/N
g15 = 21 x 10-3 Vm/N
There are two types of piezoelectric constants: the piezoelectric strain constants (d)
and the coefficient of voltage output (g).
(5a) Piezoelectric strain constant
This is a measure of the strain that occurs when a specified electric field is applied to
a PZT material. A formula to obtain the piezoelectric strain constant is as follows:
(AE0505D16)
(AE0505D16)
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16
d= kE
T
Y
ε (m/V) (2.4)
where k : coefficient of electromechanical coupling,
Tε : dielectric constant,
and YE : Young’s modulus (Newton/m2).
(5b) Voltage output constant
It is defined as the intensity of the electrical field caused when a specified amount of
stress is applied to a material (under the condition of zero displacement). A formula
to obtain the voltage output constant is given below [Tokin, 1996]:
g = T
d
ε(mV/N) (2.5)
where d : piezoelectric strain constants (m/V), and
Tε : dielectric constant.
(6) Poisson’s ratio
Poisson’s ratio for AE0505D16 is 0.34
(7) Temperature coefficient
Tk(fr) for -20 to 20o C = 200 (parts/million/oC)
Tk(fr) for 20 to 60o C = 900 (parts/million/oC)
Tk(oC) for -20 to 20o C = 3800 (parts/million/oC)
Tk(oC) for 20 to 60o C = 3500 (parts/million/oC)
(AE0505D16)
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17
The temperature coefficient is a measure of the variation of the resonant frequency
and the static capacitance with change in temperature. The formulas to obtain the
voltage output constant are given below [Tokin, 1996]:
Tk(f)= )/(10)()(1 6
20
21 CPPmxf
tftf
t°−
∆ (2.6)
Tk(C)= )/(10)()(1 6
20
21 CPPmxC
tCtC
t°−
∆ (2.7)
where Tk(f) : Temperature coefficient of resonant frequency (PPm/oC),
f(t1) : Resonant frequency at temperature t1oC (Hz),
f(t2) : Resonant frequency at temperature t2oC (Hz),
f20 : Resonant frequency at temperature 20oC (Hz),
Tk(C) : Temperature coefficient of static capacitance (PPm/oC),
C (t1) : Static capacitance (F) at temperature t1oC,
C (t2) : Static capacitance (F) at temperature t2oC,
C20 : Static capacitance at 20oC (F),
and t∆ : Temperature difference (t2-t1) (oC).
(8) Aging rate
For the PZT actuator (AE0505D16), the aging rate (AR) for the resonant frequency
and the static capacitance, (%/10 years) are 0.5 and -5, respectively.
The aging rate is an index of the change in resonant frequency and static capacitance
with age. To calculate this rate, after polarization the electrodes of transducer are
connected together, and are heated for specific period of time. Measurements are
taken of the resonant frequency and static capacitance every 2n (at 1, 2, 4 and 8) days.
The aging rate is calculated with:
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18
(AR)=1
12
12 loglog
1
Xt
XtXt
tt
−−
(2.8)
where (AR) : aging rate for resonant frequency or static capacitance,
t1, t2 : number of days aged after polarization,
and Xt1, Xt2 : resonant frequency or static capacitance at t1 and t2 days
after polarization.
(9) Mechanical quality factor (Qm)
For the AE0505D16, the mechanical quality factor (Qm) is 70. The formula to obtain
mechanical quality factor (Qm) is given below [Tokin, 1996]:
Qm=)(2 22
2
rarr ffCZf
fa
−π (2.9)
where fr : resonant frequency (Hz),
fa : antiresonant frequency (Hz),
Zr : resonant resistance (Ω),
and C : static capacitance (C).
Applications based on the piezoelectric resonance, e.g., resonators, require high
mechanical quality (Qm).
(10) Curie temperature
For the AE0505D16, the Curie temperature is 145oC. This is the temperature at
which polarization disappears (the piezoelectric qualities are lost); see also the
previous discussion.
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19
(11) Density
For the AE0505D16, the density is 8000 kg/m3. A formula to calculate the density is
given below [Tokin, 1996]:
D= V
W(kg/m3) (2.10)
where W : mass (kg) of ceramic material,
and V : volume (m3) of material.
2.2.3 PZT Actuator Manufactur ing and Operation [Setter , 2002]
The piezoelectric material’s properties can be tailored to the system’s requirements
by controlling the actuator’s chemical composition and the fabrication process of
piezoelectric actuators. In the beginning of the development process of piezoelectric
actuator, sheets of the piezoelectric material are chosen from the standardized
material types and there is a dialog between the manufacturer and the user. Then, the
chosen sheets of piezoelectric material are inspected in order to suit the specific
requirements. During this process, the manufacturer might add a number of certain
substances in order to increase the specific features of piezoelectric material
properties such as an increase dielectric constant, control conductivity, and an
increase the piezoelectric coefficients. Next, the holes pattern shown in Fig. 2.5 is
punched into the sheets.
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20
Figure 2.5 The holes pattern for manufacturing sheets.
The holes and electrode areas on a piezoelectric layer provide mechanical and
electrical connections among stacked identical layers. Next, the inspected and
punched sheets are pressed and burned (so-called the sintering process) at a certain
temperature and a certain pressure, to form a coherent mass (see Fig. 2.6). The
sintering temperature and pressure vary, as they depend on the chosen standardized
material. With such a sintering technique, the thickness of one ceramic layer (mainly
containing Ag-Pd alloy) can be reduced to less than 110 µm, thereby resulting in a
compact multilayer piezoelectric actuator. Later, the stacked and burned ceramic
layers are then patterned on and coated by the green sheet. The piezoelectric actuator
is retested to verify the adequacy of the mechanical output as a function of an applied
DC voltage. Fig. 2.6 presents the final product of the multilayer piezoelectric
actuator.
TOP VIEW OF
ELECTRODE AREAS
Edge line
× Polarity vector
Polarity vector
Gray indicates negative charge
White indicates positive charge
Typical hole for conductive path between film layers
Typical hole for mechanical and electrical connection
BOTTOM VIEW OF
ELECTRODE AREAS
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21
Figure 2.6 Stacked layers of piezoelectric actuator AE0505D16 [Tokin, 1996].
The piezoelectric actuators also have several advantages including large generated
force (AE0505D16 = 850 N), fast response speed (AE0505D16 = 22.8 KHz),
nanometers accurate positioning, compact (AE0505D16 = 1/10 the volume of a
conventional multilayer actuator), and low cost. However, the piezoelectric actuator
also entails several disadvantages, such as its poor ability in receiving tension,
flexing and twisting type of loads. To prevent the load conditions from occurring, the
prestress technique is the most commonly recommended by manufacturers. The
piezoelectric actuator also has limited operating voltages and stroke that also
influences the overall mechanism’s work range. The maximum drive DC voltage for
AE0505D16 is 150 volts, but the recommended drive is 100 volts. The displacement
of AE0505D16 resulting from the maximum drive voltage is 17.4 ± 2 microns,
while that resulting from applying the recommended drive voltage is 11.6 ± 2
microns.
2.2.4 Modeling and analysis of PZT devices
PZT devices (actuators and sensors) contain multi-domains of sciences and
engineering. This has resulted in diverse standardized terminology, which has
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22
hindered an efficient development of design knowledge for these devices. Several
efforts have been made to unify the terminology of PZT devices: see Standards
committee and Piezoelectric Crystals committee [1949] for material properties,
Committee on Piezoelectric and Ferroelectric Crystals [1958] for measurement of the
properties, and IEEE [1978] for the properties, concepts, and measurements. Mason
and Jaffe [1954] compared several methods of measuring the piezoelectric material
properties; in particular, the piezoelectric, dielectric and elastic coefficients of
crystals. Such studies are believed to have a positive impact to the standard
development.
The modeling of PZT devices usually goes along with numerical methods such as
finite element method (FEM). Alik and Hughes [1970] discussed a finite element
formulation for a single PZT device based on the variation principle. These authors
appear to have laid down a foundation for ANSYS.
Lerch [1990] used a finite element method to perform a vibration analysis of the
piezoelectric parallelepiped piezoceramic. Specifically, the author used a dedicated
FEM package which was developed to model the piezoelectric effect. The author
compared the simulation result with measurements and obtained errors from 5 % to
30 %. Peelamedu et al. [2001] studied several different scenarios of PZT devices in
order to verify that their finite element code is versatile. In the finite element model,
the base of specimen of the piezoelectric PZT-4 is constrained to be in contact with
the XY plane to eliminate the rigid body motion in the X and Y direction. Such an
approach to constrain the PZT could suffer from several problems: (1) impeding the
understanding of the actual response of the PZT, which becomes very sensitive to the
precise control of the PZT device behavior when the motion range of a PZT actuation
is very small (in micron), and (2) introducing a constraint that might be difficult to
realize in the real application situation, where the PZT device drives another
mechanism. One approach is to use glue, which might, however, create some
unwanted tension in the PZT device. The pre-stress approach is usually
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23
recommended for this purpose. It is noted that the results produced by Peelamedu et
al. [2001] remain to be verified.
There have been many other studies on the finite element modeling of PZT devices
for various applications: for example [Kim et al., 1999] for noise rejection, [Preissig
and Kim, 2000] for a piezoelectric bending actuator, [Piefort and Preumont, 2000] for
a bimorph PZT actuator using an element type called bimorph beam, and [Cattafesta
et al., 2000] for piezoelectric actuators in active flow control systems.
In [Kim et al., 1999], there was no mention of the rationales behind choosing those
particular elements or whether this work had investigated several different mesh
densities prior to determining this particular type of mesh density. The results remain
to be experimentally verified. In [Preissig and Kim, 2000], there was no mention
about the use of the manufacturer data of the piezoelectric bending actuator, and the
necessity to transfer the published data into ANSYS format (which is found
necessary; see later discussion in this thesis). In [Piefort and Preumont, 2000], there
was no mention about the rationales of using those certain mesh densities as
well as the element properties of the bimorph beam. The verification of the
theoretical data with the real motion of bimorph beam might also need to be
presented in order to more adequately understand the real behavior of the bimorph
beam.
Last, in [Cattafesta et al., 2000], the chosen finite elements to model the system were
not illustrated. The comparison between the experiment and the FEM shows some
disagreement. The authors argued that a likely cause for the observed discrepancies
between the theory and the experiment is an over-simplification of the bonding layer.
Thus, the FEM may need to be modified to model the shear deformation in the
bonding layer.
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2.3 Compliant Mechanisms
Compliant mechanisms are devices used to transfer or transform motion, force and
energy by use of the deflection of its members [Howell, 2001]. Unlike rigid link
mechanisms, compliant mechanisms gain their mobility from the deflection of
flexible members rather than from movable joints. Because compliant mechanisms
gain their mobility from the deflection of flexible members rather than from movable
joints, the required total number of components in the compliant mechanism is
significantly reduced. This enables compliant mechanisms to be manufactured as a
single piece. An example of the compliant mechanism discussed in this thesis work,
is illustrated in Fig. 2.7. The motion of a single piece prevents assembly errors and
also some inherent problems with the rigid joint (backlash and frictions) from
occurring. These advantages motivate certain applications to employ compliant
mechanisms, particularly applications that require an accurate and stable operation
such as the cell manipulation system discussed in this thesis work.
The concept of compliant mechanisms has existed for millennia. Archaeological
evidence suggests that bows (one of the earliest examples of compliant mechanisms)
have been in use since 8000 B.C [McEwen et al., 1991]. Catapults are an example of
the use of compliant mechanisms as early as the fourth century B.C. [De Camp,
1974]. At present, compliant mechanisms have found numerous applications from
daily use objects (such as skateboards, computer joysticks, and door hinges) to more
sophisticated systems (such as surgery automation in medical devices, chip assembly
in the semiconductor industry, and cell manipulation in biotechnology).
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25
Despite many of these advantages, compliant mechanisms have several
disadvantages, specifically; the flexure hinges of compliant mechanisms have certain
limitations. First, the flexure hinges have a limited range of motion in the desired axis
of rotation, whereas the conventional revolute joints have an infinite range of motion
in the desired axis of rotation as illustrated in Fig. 2.8. Consequently, the mechanisms
that employ revolute joints may have a larger work range compared to those
that employ flexure hinges.
Second, unlike the revolute joints, the flexure hinges are not fully fixed in all
directions of loading except at the desired axis of rotation. Thus, flexure hinges will
twist when subjected to torsional loads and exhibit shear deformation when subjected
to shear loads. Last, a compliant mechanism could easily induce the fatigue problem
Figure 2.7 Manufactured compliant main body [Zou, 2000].
Figure 2.8 Revolute joint of rigid body versus compliant body [Zou, 2000].
Revolute joint type in r igid body Revolute joint type in compliant body
(Flexure hinges)
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26
because its operation relies on the deformation of the material especially repeated
deformations.
In [Lorenz et al., 1990], a compliant fingertip sensor is presented (Fig. 2.9). Such a
sensor was intended for use in grippers where force feedback information was
needed. The compliant mechanism was made up of room temperature vulcanizing
(RTV) silicone rubber (see Fig. 2.9). The PZT sensor was made up of polyvinylidene
fluoride (PVDF) film. There were four strips of such PZT sensors (or films) pasted
on the compliant mechanism. This sensor could detect normal force, two tangential
force components, and torque about the normal axis. The deformation in the RTV
body occurs when a force is applied to the finger tip sensor. Next, this deformation is
transferred to the piezoelectric film materials. The shift in electrical charge in the
strained piezoelectric film is the signal used to measure the forces applied to the
sensor. Each different component of the force applied to the sensor (whether it is
normal, tangential, or torque) will produce a unique signal in each of the four pieces
of the piezoelectric film. After the signals have been amplified, they are sent to a
computer for decoupling (which translates the applied force into its independent
components). In their work, finite element analysis was used to determine the optimal
shape of the fingertip and the location/size of the PVDF film piezoelectric sensing
element. However, there was no mention in this paper regarding the particular finite
element commercial software that was used, the procedure to perform the finite
element model of the RTV body, the piezoelectric film, and the modeling interaction
between the piezoelectric film and the RTV body does not follow. The procedure in
attaching the piezo elements (PVDF) onto the compliant mechanism (RTV) was
organized into three steps. First, it was required to form the rubber into the correct
fingertip shape. Such process was accomplished by pouring the liquid rubber into a
mold. Second, the piezo elements (PVDF) were cut with a good-quality scissors. Yet,
because the fact that the PVDF is anisotropic by nature; care must be taken to cut the
film in the proper direction (not explained further). To complete the process, the wire
leads were added by use of a conductive epoxy. Third, a primer was used to bond the
piezoelectric film to the rubber. A primer is a chemical additive that simultaneously
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bonds to the film and vulcanizes the rubber. Finally, the authors compared the
simulation and experimental results for the sensitivity ratio (the sensitivity ratio was
defined as the ratio of length of the major axis of the ellipse to its minor axis
corresponding to the largest amplitude of the signal for a given force). The
experimental results were greater than the simulation results by 25%.
Several other studies on finite element modeling for the PZT compliant mechanism
may be noticed, e.g., [Angelino and Washington, 2002; Abdalla et al. 2003; Chen and
Lin, 2003; Bharti and Frecker, 2004]. Among these works, only Bharti and Frecker
[2004] provided a reasonably detailed discussion of the finite element modeling. The
authors used three PZT actuators and a compliant mechanism to develop a stabilized
rifle mechanism. Such a mechanism stabilizes the rifle position by removing error
sources (the undesired movement of the barrel resulting from extreme psychological
stress experienced by a soldier during combat). The actuators compensate for the
small undesired motions of the barrel, thereby stabilizing the barrel assembly. The
objective of this work was to predict an optimal compliant mechanism design
surrounding the PZT actuator with maximum stroke amplification. The authors used
Figure 2.9 Finger tip sensor.
Signal amplifiers device
Basic RTV body
PVDF piezo elements
Forces applied to sensor
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commercial software called ProMechanica. A main body was made of Aluminum
7075, while the employed piezoelectric actuator was PZ26. In their experiment, the
stack actuators were preloaded by press fitting them into the compliant mechanism.
Equal preload on each actuator was assured by previously measuring the voltage
change due to the compressive preload. In the finite element model however, the
connection between the piezoelectric element and the compliant mechanism was not
discussed. In addition, an equivalent temperature change was applied to the
piezoelectric. It seems that the model was not completely inclusive in the finite
element model. In particular, a customized code which computes the voltage from the
temperature was needed and integrated with the rest of the finite element model.
2.4 Finite Element Analysis by Use of ANSYS
ANSYS is a finite element software package that was first commercially available in
1970 (Swanson Analysis Systems, Inc.). Since then, ANSYS has been used by design
engineers throughout the world for such engineering applications as structural,
thermal, fluid, and electrical analyses. In this thesis work, ANSYS was used as a
computational tool for modeling the RRR mechanism. It is noted that the RRR
mechanism basically consists of a compliant main body and three PZT actuators (see
Fig. 2.10).
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In ANSYS, there are five typical steps for performing a finite element analysis as
illustrated in Fig. 2.11. The first step is to gather the data of the problem. Such data
may be available in forms of engineering drawings on paper, data specifications from
manufacturer, or conceptual design. The second step is to build a finite element
model for the application problem. This step consists of such activities as defining
units, selecting types of elements, defining material properties, and creating the finite
element model. As for defining a system of units, it should be noted that the ANSYS
program does not assume a system of units. Thus, the users are responsible to
maintain the consistency of system of units for all the input data in the ANSYS
program. As for selecting element types, the decision is based on the characteristics
of element type to best model that application problem geometrically and physically.
The material properties are required for most element types. Depending on the
element types, material properties may be linear or non-linear; isotropic, orthotropic,
or anisotropic; and constant temperature-independent or temperature-dependent.
Bolt
Bolt Bolt
PZT
End-effector platform
Main body
Figure 2.10 RRR mechanism.
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There are two methods to create a finite element model in ANSYS: automatic
meshing (also called the solid modeling in ANSYS terminology) and manual
meshing (also called the direct generation in ANSYS terminology). In automatic
meshing, users are required to have a solid model available prior to the creation of a
finite element model. When such a solid model becomes available, the users then can
instruct ANSYS to automatically develop a finite element model (nodes and
elements). The purpose of using automatic meshing is to relieve the user of the time-
consuming task of building a complicated finite element model. However, this
method requires significant amounts of CPU time and sometimes fails to maintain the
connectivity of nodes and elements. In manual meshing, the users are to define the
nodes and the elements directly (development of a solid model is not required). The
manual meshing method offers a complete control over the geometry and
connectivity of every node and every element, as well as, the ease of keeping track of
the identities of nodes and elements. However, this method may not be as convenient
as the automatic meshing when dealing with a complicated finite element model. It is
possible to combine both methods.
The third step is to build a solution. This step includes such activities as applying
loads, selecting boundary conditions, and selecting types of analysis. The loads are
defined in several disciplines such as structural (displacements and forces), thermal
(temperatures and heat flow rates), electrical (electric potentials and electric current)
and fluid (velocity and pressure). In terms of region of where the loads are applied,
loads can be classified as a nodal load (a concentrated load applied at a node in the
model such as forces and moments in structure), a surface load (a distributed load
applied over a surface such as pressures in fluid), and a body load (a volumetric load
such as heat generation rates in thermal analysis). The fourth and fifth steps could be
achieved with some sufficient understanding of the finite element software and the
real system respectively.
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Figure 2.11 General procedures to perform FEA by use of ANSYS.
2.5 The Natural Frequency of the Compliant Mechanism
By definition [Braun et al., 2001], the natural frequency of a system describes the
individual ways in which the system will choose to vibrate without any external
applied excitation other than natural disturbances (such as gravitational force,
2. BUILD MODEL i. Define units ii. Select element types iii.Define material properties iv.Create finite element model
3. BUILD SOLUTION i. Apply loads ii. Select type of analysis iii. Solve the problem
4. REVIEW RESULTS i. Graphical presentation ii. Values presentation
1. GATHER REAL PROBLEM’S DATA i. Engineering drawings ii. Manufacturer’s specification iii. Conceptual design
5. RESULTS’ INTERPRETATION
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centrifugal force, and elastic restoring force). The natural frequency is positively
correlated to the stiffness of a system. The higher natural frequency means the higher
stiffness. Therefore, the natural frequency is a measure of the stiffness. For the
mechanism, there are sets of configurations. At each configuration, the mechanism is
like a structure with its degree of freedom (DOF) being zero. The natural frequency
for a mechanism is then calculated at each configuration.
Kitis and Lindenberg [1989] used the transfer matrix method, as an alternative of the
finite element method, to compute the natural frequencies of the four-bar mechanism.
By use of the transfer matrix, the mechanism was modeled as a combination of
massless beam sections, while the lump masses can be calculated through successive
multiplication of point and field matrices along the link. To calculate the overall link
transfer matrices, it is required to develop a transfer matrix to relate the state vectors
of the adjoining links (pin joint transfer matrix). In the transfer matrix approach, the
size of the system matrix is reduced. The results from the transfer matrix method
were compared to those from the finite element approach from a different work study
[Turcic, 1982]. The comparison results indicated considerably agreement.
Jen and Johnson [1991] calculated the natural frequencies of a planar robot, in
particular three-link manipulator. The authors also studied the effect of variations of
the physical parameters on the natural frequencies, by use of the component mode
synthesis (CMS) approach. The CMS approach basically disassembles a complete
structure into substructures and then computes the mode for each component. The
corresponding mass matrix and stiffness matrix are then derived for each component
and subsequently “assembled” by use of the displacement compatibility conditions at
the component interfaces. The system was modeled by three beam elements
connected by two stiff revolute joints. All the computations were performed by use of
a general purpose language package (MATLAB in particular) without using special
purpose finite element packages. The obtained results have not been verified with
experimental results.
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Li and Sankar [1992] focused on the development of a procedure to derive dynamic
equations of motion for flexible robot manipulators. The derived dynamic equations
of motion would facilitate the computation of the manipulator’s behaviors
(particularly the position and velocity of first mode and second mode, the joint angle
position and velocity, and the joint actuator torque) with regard to the elapsing time.
The procedure consisted of the development of kinematics of flexible links,
lagrangian equations of motion for flexible manipulators (kinetic energy and
potential energy of flexible links, development of flexible manipulator equations of
motion). That method has been verified by use of computer simulation from other
papers. The authors claimed that the method proposed in their paper was simple,
more systematic, and efficient. It should be noted, however, this method is relatively
simpler as it deals with a single-link robot manipulator.
Iwatsuki et al. [1996] proposed a new approach to study the vibration behavior of
spatial serial manipulators composed of multiple elastic links. This method was used
to calculate the internal forces and moments interactively acting between the two
adjacent links connected with joint. The calculated results have been validated with
the experimental results for various motions. Such an approach may be applied more
effectively to the system with few joints. However, for the parallel manipulators that
might have large numbers of joints, this approach becomes difficult to use due to the
complex nature of the approach.
Lyon et al. [1999] proposed the pseudo-rigid-body model (PRBM) approach to
predict the first modal frequency of compliant mechanisms. Their approach was
verified by the experimental set-up approach (by use of digital oscilloscope). The
results of the two theoretical approaches showed good agreement (within 9 %
deviation) with the experimental results.
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34
2.6 The Stiffness of the Compliant Mechanism
Several earlier studies have shown that accurate control and large work range are
related to the stiffness and the natural frequency of the micromanipulation system.
Han et al. [1989] proposed a procedure to optimize a 6 degree-of-freedom (6-DOF)
fully-parallel micromanipulator for enhanced accuracy. They observed a need of
trade-off between large work range and control accuracy through the stiffness
property of a micromanipulator. Tomita et al. [1992] proposed a method of
determining the design of a ultra precision stage (for the semiconductor
manufacturing application) using a parallel linkage mechanism. As well, the authors
discussed the necessity of high displacement resolution and high frequency response
to compensate for the vibrating disturbances of the environment. Sanger et al. [2000]
explained that the accuracy of a manipulator particularly under different loads is
directly related to its stiffness, and that knowledge of the stiffness can be used to
develop a means of simultaneously controlling the force and displacement for a
partially constrained end-effector. Portman et al. [2000] proposed a new structural
concept for a type of closed kinematic chain mechanism, (e.g., a 6 x 6 parallel
platform mechanism). This new concept involved the application of welded joints.
The objective of this structural concept is to obtain high stiffness and high accuracy.
The stiffness of a mechanism is also related to the so-called singularity posture of the
mechanism [Gosselin, 1990].
There are generally two kinds of methods available to model the system stiffness.
The first method is the structural analysis method in which the system stiffness is
directly associated with the number of nodes of elements that model a structure ( a
mechanism at a particular configuration). The second method considers the
relationship between the force (including the moment) and the displacement
(including the angular displacement) at the end-effector. In literature, such a stiffness
may be called global stiffness. Gosselin [1990] presented a method for calculating the
global system stiffness, which results in Eqn. (2.13).
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35
[K] = k JT J (2.13)
where: [K] : the global stiffness matrix,
k : the stiffness along the actuator axis,
JT : the transpose of Jacobian matrix of the mechanism,
and J : the Jacobian matrix of the mechanism.
Zhang and Gosseline [1999] went on to develop a similar formula as Eqn. (2.13),
with inclusion of the stiffness of each link component. El-Khasawneh and Ferreira
[1999] further studied the maximum and minimum stiffness, as well as, their
orientation. In their study, they defined the so-called general stiffness.
ppS
T
T
∆∆= ττ
(2.14)
where: S : the general stiffness matrix,
∆p : the position and orientation of the end-effector,
and τ : the required input to cause the platform to experience ∆p.
Also,
τ =kJTJ∆p (2.15)
The eigenvalue of JTJ can be found, assuming λmin = λ1 ≤ λ2 ≤ … ≤ λ6 = λmax. Then
they found
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36
K λmin ≤ S ≤ K λmax (2.16)
The direction of λmin (λmax) corresponds to the normalized eigenvectors corresponding
to λmin (λmax). It is noted that their method has not considered the stiffness of the link
and has assumed that all the actuators have the same axial stiffness.
2.7 Concluding Remark
The compliant mechanism is a very promising concept to build the micro-motion
device. The conventional approach to modeling a compliant mechanism is the pseudo
rigid body method. There are two problems with this method. First, the method can
not capture the whole material distribution in the compliant mechanism domain.
Second, the dynamic model with this method is extremely complex and lengthy
(despite its analytic form) as shown by Zou [2000], which can subsequently prohibit
any exploration of the dynamic model for the real-time control of a compliant
mechanism. The finite element method is definitely a useful tool for the analysis of
compliant mechanisms. However, the use of general-purpose finite element methods
for (1) motion analysis with consideration of couplings of the PZT actuator and the
compliant material and (2) natural frequency and stiffness analysis warrants further
study.
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37
CHAPTER 3
FINITE ELEMENT ANALYSIS OF
DISPLACEMENT OF RRR MECHANISM
3.1 Introduction
This chapter presents a study of finite element analysis of the RRR mechanism by
making use of the finite element commercial software called ANSYS. Zou [2000]
previously conducted finite element analysis (FEA) for the RRR mechanism with
some limitations. This thesis work is expected to overcome these limitations;
specifically by including the PZT actuator in the FEM model. The organization of
this chapter is as follows: Section 3.2 will present some fundamental information that
is necessary to facilitate discussions in this chapter. Section 3.3 will present a model
for the kinematic analysis of the RRR mechanism. Section 3.4 will present finite
element modeling of the PZT actuator. Section 3.5 will discuss a procedure to
incorporate PZT into the RRR mechanism. Section 3.6 illustrates how the model
works by using an example. Section 3.7 presents a summary with some discussion.
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38
3.2 Basic Information of ANSYS
ANSYS provides several new types of elements to model the piezoelectric effects, or
in general to model those effects that are related to domains of disciplines, e.g.,
electrical-pressure, electrical-thermal, etc. In this section, a type of element for
modeling the piezoelectric effect will be presented.
3.2.1 Multidisciplinary Element Type [ANSYS, 2004]
Multidisciplinary element types are used to capture the effects that are related to two
different domains of disciplines, e.g., electrical-pressure, electrical-thermal, etc. In
this section, the type of element for modeling the piezoelectric effect will be
presented. The PZT actuator system has electrical behavior (the electrical current as
input to the PZT actuator) and mechanical behavior (the existence of PZT actuator’s
deformation as the output for the PZT actuator). Finite elements must capture this
mechanical-electrical joint behavior. In ANSYS, there are two types of elements for
modeling the piezoelectric effect, namely SOLID 5 and PLANE 13.
• SOLID 5
Figure 3.1 Geometry of SOLID 5 [ANSYS, 2004].
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DISCIPLINE DEGREES OF FREEDOM ACTIVATION
Coupling of structural,
thermal,
electrical and
magnetic
UX, UY, UZ, TEMP, VOLT,
MAG
KEYOPT (1) =0
Coupling of thermal,
electrical and
magnetic
TEMP, VOLT, MAG KEYOPT (1) =1
Structural UX, UY, UZ KEYOPT (1) =2
Coupling of structural and
electrical, also
called as
piezoelectric
UX, UY, UZ, VOLT KEYOPT (1) =3
Thermal TEMP KEYOPT (1) =8
Electrical VOLT KEYOPT (1) =9
Magnetic MAG KEYOPT (1) =10
SOLID 5 is a type of element that occupies three-dimensional space. It has eight
nodes. Each node has three displacements along the x, y, and z axis, respectively. A
prism-shaped element is formed by defining duplicate node numbers as described in
Figure 3.2 Disciplines in SOLID 5 [ANSYS, 2004].
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40
Fig. 3.1. In particular, one can define a prism-shaped element by defining nodes K, L
and nodes O, P in same locations, respectively. The prism-shaped element may be
useful in modeling a system that has a geometric curvature (e.g., cylinder). In this
thesis work, the brick-shaped element is chosen due to the fact that the geometrical
shape of the piezoelectric actuator does not have any curvature.
The SOLID 5 element is capable of modeling seven different types of disciplines (see
Fig. 3.2). The meaning of these terms in the second column in Fig. 3.2 is presented in
Fig. 3.3. The third column in Fig. 3.2 is used in the ANSYS code to select one
particular discipline. For the discipline corresponding to the problem discussed in this
thesis, KEYOPT (1) =3 is chosen. When this particular type of discipline is chosen,
ANSYS will only consider (compute) the behaviors of SOLID 5 in UX, UY, UZ and
VOLT degrees of freedom. It should be noted that UX, UY and UZ are to indicate
the displacements in the X, Y and Z directions (X, Y and Z axes are based on the
global coordinate system), while VOLT is to indicate the difference in potential
energy of the electrical particles between two locations.
• PLANE 13
PLANE 13 is a type of element that occupies the two-dimensional space. It has four
nodes. Each node has two displacements along the X and Y axes respectively. A
triangle-shape element can be formed by defining node K and node L in a same
Figure 3.3 Geometry of PLANE 13 [ANSYS, 2004].
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location. Triangle-shape element is more adaptable to complex shapes of the object.
Due to the regularity in shape with a real PZT actuator, the quadrilateral-shape
element was chosen to model the piezoelectric actuator in the two-dimensional space.
For the problem under study in this thesis, KEYOPT (1) = 7 (see Fig. 3.4) was
chosen.
DISCIPLINE DEGREES OF
FREEDOM
ACTIVATION
Magnetic AZ KEYOPT (1) =0
Thermal TEMP KEYOPT (1) =2
Structural UX, UY KEYOPT (1) =3
Coupling of structural, thermal
and magnetic
UX, UY, TEMP, AZ KEYOPT (1) =4
Thermal and Magnetic VOLT, AZ KEYOPT (1) =6
Coupling of structural and
electrical
UX, UY, VOLT KEYOPT (1) =7
Figure 3.4 Disciplines in PLANE 13.
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DEGREES OF FREEDOM MEANING
UX Translation in the X direction
UY Translation in the Y direction
UZ Translation in the Z direction
TEMP Temperature
VOLT Electric potential (source current)
MAG Scalar magnetic potential
AZ Z-component of vector magnetic potential
3.2.2 Piezoelectr ic mater ial data
• Manufacturer data versus ANSYS data
Section 2.2.2 presented and discussed the entire data specification of the PZT
actuator (in particular, those material properties for the AE0505D16 model from
TOKIN). However, not every single data presented in Section 2.2.2 is required in
modeling the PZT actuator with ANSYS. There are two reasons for this situation.
First, the real piezoelectric materials entail the mechanical and electrical dissipations,
strong non-linear behavior, hysterisis effects, and aging effects [IEEE, 1978].
However, these characteristics are not considered in a linear theory of piezoelectricity
Figure 3.5 Degrees of freedoms.
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by Allik and Hughes [1970]. The linear theory of piezoelectricity is a theory in which
the elastic, piezoelectric, and dielectric coefficients are treated as constants. In the
real situation, the elastic, piezoelectric and dielectric coefficients are in the form of
functions of the magnitude and frequency of applied mechanical stresses and electric
fields. Allik and Hughes [1970] laid a foundation of the mathematical procedure of
ANSYS in solving a piezoelectric material problem. Therefore, ANSYS only
considers those properties, including elastic constant matrix, permittivity constant
matrix and piezoelectric constant matrix. Second, the entire data specification of the
PZT actuator is to provide the complete measured performance of the PZT actuators
under certain testing conditions. Particular applications may only need a subset of
these conditions. Therefore, they may only need a subset of the material property
data.
Furthermore, the data specification from manufacturers cannot be directly entered
into the ANSYS program. This is because the matrix format supplied by most of the
PZT manufacturers (including TOKIN) do not have the same definition as the matrix
formats provided by ANSYS. Such a gap in the material property data between the
manufacturer and ANSYS can be further illustrated.
ANSYS requires three types of data for modeling the PZT actuator, which are the
stiffness matrix (mechanical discipline), permittivity at constant strain (electrical
discipline) and piezoelectric stress matrix (coupling-discipline between mechanical
and electrical disciplines). However, most PZT manufacturers only provide
compliance matrix (mechanical discipline), permittivity at constant stress (electrical
discipline) and piezoelectric strain matrix (coupling-discipline between mechanical
and electrical disciplines). Thus, a conversion to create the same definition is
necessary. Such a conversion is realized by a program called PIEZMAT macro
provided by ANSYS.
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• The PIEZMAT macro
The work of Allik and Hughes [1970] that underlies the mathematical procedure of
ANSYS in solving a piezoelectric problem resulted in the constitutive equations for
piezoelectricity, and such equations are represented in ANSYS as follows:
T = [cE] S - [e] E (3.1)
D = [e]T S + [εS] E (3.2)
where T : stress vector (six components x, y, z, xy, yz, xz),
S : strain vector (six components x, y, z, xy, yz, xz),
D : electric displacement vector (three components x, y, z),
E : electric field vector (three components x, y, z),
[cE] : stiffness matrix evaluated at constant electric field,
[e] : piezoelectric matrix relating stress and electric field,
[e]T : transpose of [e],
and [εS] : dielectric matrix evaluated at constant strain.
However, most of the manufacturers of piezoelectric materials publish the data
specification based upon the following equations:
S = [sE] T + [d] E (3.3)
D = [d]T T + [εT] E (3.4)
where T : stress vector (six components x, y, z, yz, xz, xy),
S : strain vector (six components x, y, z, yz, xz, xy),
D : electric displacement vector (three components x, y, z),
E : electric field vector (three components x, y, z),
[sE] : compliance matrix evaluated at constant electric field,
[d] : piezoelectric matrix relating strain and electric field,
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[d]T : transpose of [d],
and [εT] : dielectric matrix evaluated at constant stress.
ANSYS requires the data specifications (in particular piezoelectric matrix,
compliance matrix and permittivity matrix) that provide their data specifications
based on Eqns. (3.1) and (3.2). However, the PZT manufacturers are based on Eqns.
(3.3) and (3.4). In order to realize the conversion, Eqns. (3.3) and (3.4) can be
rewritten into the following forms:
S = [sE] T + [d] E from Eqn. (3.3)
[sE] T = S - [d] E
T = [sE]-1 [S] - [sE]-1 [d] E (3.5)
D = [d]T T + [εT] E from Eqn. ( 3.4)
D = [d]T [sE]-1 [S] - [sE]-1 [d] E + [εT] E
D = [d]T[sE]-1 [S] +( [εT] - [d]T sE -1[d]) E (3.6)
Comparing Eqns. 3.5 and 3.6 with Eqns. 3.1 and 3.2, respectively, results in the
following relations.
[cE]= [sE]-1 (3.7)
[εS]= [εT] - [d]T sE -1[d] (3.8)
[e]= sE -1[d] = [d]T[sE]-1 (3.9)
3.3 Kinematic Analysis of the RRR Mechanism
Zou [2000] presented a preliminary study on the kinematic analysis of the RRR
mechanism using ANSYS. The model is parametric in the sense that a set of
kinematic parameters govern the model. The model used the quadrilateral element
type. This model suffers from the following defects: First, there are some poor
shaped elements (see Fig. 3.6). This point was also observed by Zettl [2003]. Second,
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46
the PZT actuator is not considered in the model. Specifically time-dependent
prescribed motions at three actuators are implemented by giving the time-dependent
nodal displacement. This has introduced a considerable approximation with respect to
the real situation. The next several sub-sections will discuss how these limitations are
overcome to result in a better model.
Figure 3.6 Zou’s Finite Element Model of the RRR mechanism.
Bolt 1
Bolt 2
Bolt 3
Triangle Platform (
Main body
End-effector
Quadrilateral elements
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47
Note that the physical prototype of the RRR mechanism can be found in Appendix A.
Fig. 3.7 illustrates the current finite element model of the RRR mechanism. The
Figure 3.7 Finite Element Model of the RRR mechanism in this thesis.
Bolt 1
Bolt 2
Bolt 3
End-effector platform
PZT 1
PZT 2
PZT 3
Thin metal plate 1
Thin metal plate 2
End-effector
Triangular elements
Application of zero translations
Thin metal plate 3
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model takes the piezoelectric actuators behavior into consideration. The triangular
elements with 6 nodes (Fig. 3.7) replace the previous quadrilateral elements with 8
nodes (Fig. 3.6). By considering the fact that the shape of the actual end-effector
platform in the experiment is circular, hence, the shape of the end-effector platform
in the current finite element model is also modeled to be circular.
To avoid the three-dimensional analysis that requires tedious and large numerical
effort, the RRR mechanism was modeled by two-dimensional finite elements. For the
two-dimensional finite element, one needs to determine whether to apply the plane
stress or the plain strain model. Zou [2000] used the plane stress model for the RRR
mechanism by arguing the depth of the RRR mechanism was considerably thin. Zettl
[2003] observed that in the region of the flexural hinge, the plane stress model was
not a proper choice based on a three-dimensional finite element analysis conducted
by him. This observation does imply a finite element model which takes the plane
stress for most of the regions of the compliant mechanism and the plane strain for the
region of the flexure hinge. However, the physical setting upon which Zettl [2003]
made his observation, is an “ isolated” flexural hinge in the sense he considered the
material other than the flexure hinge to be rigid. In the real mechanism, the flexural
hinge is somewhat “merged” in a relatively large main body that deforms
substantially, and that material is very thin. Therefore, the plane strain behavior in a
small region may be constrained by the plane stress behavior in a relatively large
region, which is a speculation. In this thesis work, the plane stress model was applied
for the whole region of the material except for the PZT actuator which was modeled
with the plane strain model. The plane strain model was chosen to model the two-
dimensional PZT actuator because its deformation results were closer to the
deformation results of the three-dimensional PZT actuator in the finite element
model. Such a treatment may help examine the speculation, as raised before.
Fig. 3.8 is to facilitate explanation of the motion nature of the RRR mechanism. In
the current finite element model, one may input directly the electrical voltages on the
PZT actuators in the so-called parametric PZT loading constants. The parametric
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49
loading constants are the constants that are placed in the beginning of the developed
ANSYS codes such that the user may vary or change the loading conditions of the
PZT actuators without any difficulty. The location of the nodes, in which the
electrical voltages are applied, is given in the following example. If one wants to
apply (100, 80, 0) volts onto (PZT 1, PZT 2 and PZT 3), the locations of the nodes, in
which the electrical voltages are applied, are given in Fig. 3.9. It should be noted that
the indicated nodes in Fig. 3.9 belong to the piezoelectric element.
Figure 3.8 The motion nature of the RRR mechanism.
Electrical input of PZT
Input loading for the main body
Mechanical output of PZT
Motion amplification
End-effector
A system configuration of the RRR mechanism
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Due to its coupling-field nature, the PZT converts the applied electrical current into
mechanical deformation (the mechanical output of PZT; see Fig. 3.8). The
mechanical deformations of the PZT actuators then push the material in the direction
of the PZT actuators’ deformations. The specially designed notches and holes on the
material amplify these deformations during the process of transferring the actuator’s
deformations onto the displacement of the end-effector platform (the circular plate as
illustrated in Fig. 3.7). It should be noted that the end-effector platform is connected
to the material by use of bolts. After the deformations from the PZT actuators have
been completely transferred, the whole system will be at a rest position. An achieved
position and orientation of the end-effector at rest due to the electrical inputs of the
PZT actuators (PZT 1, PZT 2 and PZT 3) are defined as one system configuration.
Therefore, to achieve a different system configuration, one uses different values of
Figure 3.9 The application of electrical input on the PZT actuators.
PZT 3
PZT 2
PZT 1
100 volts 0 volt
0 volt
80 volt
0 volt
0 volt
Holes to attach the main body with the end-effector platform
Holes to constrain the main body onto the ground
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the PZT actuators (PZT 1, PZT 2, and PZT 3). Because there are numerous possible
different values of the PZT actuators, consequently the RRR mechanism also has
numerous system configurations. In this way, one sees a series of changes of
configurations, which is translated to the kinematic motions from a view point of
rigid body mechanisms.
3.4 Modeling of the PZT Actuator for the RRR Mechanism
For the RRR mechanism driven by the PZT actuator, the modeling of the PZT
actuator is described as follows:
Step 1. Choose a suitable type of finite element
The RRR mechanism, i.e., the compliant material, is considered as a planar finite
element problem. Furthermore, the plane stress model was considered over the whole
material region. There could be some errors produced due to such a treatment (i.e.,
the planar finite element problem). However, the error produced at the end-effector is
relatively small in comparison with the measured result (see Chapter 5 for the
deformation results at the end-effector). To incorporate the finite element model of
the PZT actuator into that of the RRR mechanism (without the PZT element), two
finite element models must be consistent. In this connection, element type (PLANE
13) is chosen for modeling the PZT actuator.
Fig. 3.10 shows the geometric boundaries of the actual piezoelectric actuator. The
piezoelectric actuator has dimensions of 5 x 5 x 20 mm. In ANSYS, the geometry of
the PZT actuator was created by use of its solid modeling.
Step 2. Build the PZT actuator in ANSYS
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a. Create a geometric model of the PZT actuator
b. Inputting the manufacturer data into ANSYS format
The modeled material properties of the PZT actuator in ANSYS consist of
mechanical matrix (compliance constant matrix), electrical matrix (permittivity
constant matrix), and mechanical-electrical matrix (piezoelectric constant matrix).
Each type of matrix is discussed as follows (note that the polarization of the PZT
actuator is in the direction of the Z-axis before working on these three-dimensional
matrices).
• Mechanical matrix (compliance constant matrix)
[ Es ] =
−
E
E
EE
E
EE
EEE
s
sss
s
ss
sss
44
44
1211
33
1311
131211
0
00)(2
1000
000
000
(3.10)
Eqn. (3.10) shows the arrangement of the manufacturer data within the ANSYS
format. Such an arrangement occurs because the manufacturer’s data has mechanical
vector in the form x, y, z, yz, xz, xy , whereas ANSYS’s mechanical vector is in the
Figure 3.10 Geometric boundaries of the PZT actuator.
5 mm
20 mm
5 mm
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form x, y, z, xy, yz, xz . [sE] is the compliance constant matrix obtained at constant
electrical field. The first subscript indicates the direction of strain, while the second
subscript indicates the direction of stress. To completely model the three-dimensional
manufacturer’s material data in ANSYS, the properties of the compliance matrix
( Es11 , Es12 , Es13 , Es33 , and Es44 ) are required. Note however that the manufacturer of the
PZT actuator in this thesis (TOKIN) only supplies two kinds of material properties
( Es11 and Es33 ). To completely model the three-dimensional manufacturer’s material
data in ANSYS, hence, the other material properties ( Es12 , Es13 , and Es44 ) are to be
computed. To facilitate computation, [sE] can also be presented in the format shown
in Eqn. (3.11).
[ Es ] = (3.11)
By comparing Eqn. (3.10) and Eqn. (3.11), Eqn. (3.12a-f) can be presented as
follows.
EX = Es11
1= EY (3.12a)
EZ = Es33
1 (3.12b)
GXY = )(2
1
1211EE ss −
= )1(2 xy
XE
υ+ (3.12c)
GYZ = Es44
1= GXZ (3.12d)
−
−−
XZ
YZ
XY
Z
Y
YZ
Y
X
XZ
X
XY
X
G
G
G
E
EE
EEE
1
01
001
0001
0001
0001
ν
νν
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XYν = -E
E
s
s
11
12 (3.12e)
YZν = -E
E
s
s
33
13 = XZν (3.12f)
where Ex : Young's modulus in the x direction,
XYν : Poisson's ratio in the X and Y directions,
GXY : shear modulus in the XY plane,
GYZ : shear modulus in the YZ plane,
and GXZ : shear modulus in the XZ plane.
Besides Es11 and Es33 , the poisson ratio of AE0505D16 ( XZν ) is given by the
manufacturer. Es13 can be found through Eqn. (3.12f). Es12 is assumed to be zero, while
Es44 can be found through Eqn. (3.12c) and Eqn. (3.12d) by assuming GXY = GYZ =
GXZ . Thus, Es44 can be expressed by )(2 1211EE ss − .
• Electrical matrix (permittivity constant matrix)
Most manufacturers (including TOKIN), presents permittivity matrix of the material
evaluated under the condition of constant stress[ ]Tε , while ANSYS requires the
permittivity matrix of the material evaluated under the condition of constant strain
[ ]Tε . The first subscript in the matrix [ ]Tε indicates the direction of the dielectric
displacement and the second subscript indicates the direction of the electrical field.
PIEZMAT macro, which is also based on Eqn. (3.8) in particular, is to perform
conversion. The permittivity constant material in Eqn. (3.13) is input into ANSYS.
[ ]
=
S
S
S
S
33
11
11
00
00
00
εε
εε (3.13)
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• Mechanical-electrical matrix (piezoelectric constant matrix)
[ ]d =
000
00000
00000
333131
15
15
ddd
d
d
(3.14)
Eqn. (3.14) shows the arrangement of the piezoelectric constant matrix from the
manufacturer’s data within the ANSYS format. The matrix [ ]d presents the
piezoelectric constant matrix that describes the relationship between strain
(mechanical properties) and a specified electrical field (electrical properties). The
first subscript indicates the direction of the electrical field, while the second subscript
indicates the direction of strain.
There are several notes that one needs to pay attention regarding modeling the PZT
actuator by using ANSYS. First, the author employed several ANSYS versions (from
ANSYS educational version 5.5 until ANSYS educational version 8.1) in modeling
the PZT actuator by using ANSYS. The author has found that ANSYS has been
updating its features in a gradual manner. In ANSYS educational version 8.1,
ANSYS has added new features and new types of elements that eliminate the need of
using the PIEZMAT macro. In particular, there are some new kinds of
multidisciplinary elements that are capable of modeling the piezoelectric actuator
directly based on the manufacturer data, i.e. PLANE 223, SOLID 226, and SOLID
227. Second, it is necessary to ensure that the test method and terminology related to
the PZT actuator have been standardized. Some manufacturers might use their own
test method. Thus, it is very important to consult with the manufacturer prior to
performing any kind of modeling regarding the defined material properties in the data
specification.
The general model of the PZT finite element model is three-dimensional with the Z-
axis as the polarization direction. For the two-dimensional PZT element (PLANE
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13), the Y-axis is the polarization direction. Therefore, there is a need of
transformation from the 3-D material property matrix to the 2-D material property
matrix. The transformation is implemented by Eqn. (3.15), Eqn. (3.16) and Eqn.
(3.17).
Figure 3.11 Axis of piezoelectric material.
[ ]
=
S
S
S
S
33
11
11
00
00
00
εε
εε
S
S
33
11
0
0
εε
(3.15)
[e] =
00
00
000
00
00
00
15
15
33
31
31
e
e
e
e
e
0
0
0
0
15
31
33
31
e
e
e
e
(3.16)
X Y Z X
Y
Z
X Y X
Y
X Y Z
X
Y
Z
XY
YZ
XZ
X Y X
Y
XY
XZ
1
2
3
4 5
6 Polarization
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57
[ Es ] =
−
E
E
EE
E
EE
EEE
s
s
ss
s
ss
sss
44
44
1211
33
1311
131211
0
00)(2
000
000
000
E
E
EE
EEE
s
s
ss
sss
44
11
1233
131211
0
0
0
(3.17)
Step 3. Mesh the PZT element
Figure 3.12 Different types of meshing density on PZT
Fig. 3.12 shows several meshing schemes for the PZT element. The objective here
was to find the least number of elements without loss of accuracy. This work
investigated twelve possible mesh configurations for various different loadings. The
ANSYS results indicated that there was no significant difference in the PZT motions
X Y Z XY YZ XZ X Y XY XZ X
Y
Z
XY
YZ
XZ
X
Y
XY
XZ
1 element
4 elements
9 elements
25 elements
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for the twelve mesh configurations. Therefore the one element mesh configuration
was chosen to model the PZT actuator.
3.5 Finite Element Modeling of the PZT RRR Mechanism
The PZT actuator and the RRR material are assembled into the RRR mechanism,
which is called the PZT RRR mechanism. There are several issues to be addressed
for modeling the PZT RRR mechanism, and they are discussed as follows.
o Model the prestress of the PZT Actuator
The effect of the prestress in finite element modeling is such that the nodes at the
interface are the subject to the extra workload. This load is calculated with the
following equation:
F = E × A × l
l∆ (3.18)
where F : the prestress force or load (N),
E : the Young modulus of the PZT material (N/m2),
A : the cross sectional area of the PZT actuator (m2),
l : the length of the PZT slot (m),
and l∆ : the displaced length of the PZT slot due to the prestress (m).
By measuring ∆l (the pre-deformation), one can find F from Eqn. (3.18). For the
RRR mechanism concerned, The pre-deformed forces of PZT 1, PZT 2, and PZT 3
are 3.981×104N, 2.56667×104N, and 5.395×104N, respectively. The detailed
procedure for measuring ∆l can be found in Appendix B.
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o Model the Thin Metal Plate
The prestress is currently implemented through inserting a metal piece between the
PZT actuator and the RRR material; details about the prestress can be found in
Appendix B. The metal piece is modeled using the element type COMBIN 14 (see
Fig. 3.13). This element is also known as a spring-damper element
Figure 3.13 COMBIN 14 [ANSYS, 2004]
The stiffness of the COMBIN 14 element can be calculated with the following
equation.
k = L
EA (3.19)
where k : the spring constant (N/m),
E : the Young modulus of the plate ( 2/ mN ),
A : the cross-sectional area of the plate (m2),
and L : the length of the thin metal plate (m).
o Model PZT within the RRR mechanism
The relationship among the PZT actuator, the metal piece, and the RRR mechanism
is illustrated in Fig. 3.14, where the extra workloads due to the prestress are also
shown, respectively.
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o Specify of the boundary conditions
All the nodes along the perimeter of A, B and C are constrained such that these nodes
do not translate (see Fig. 3.15). This is to represent the fact that the RRR mechanism
is fixed onto the ground.
o Model the bolts
The bolts E, F and G (see Fig. 3.15) in the PZT RRR mechanism fasten the main
body with the end-effector platform. The modeling should ensure that all the
corresponding elements share the same nodes on the interfaces. To ensure that the
main body and the bolts share the same nodes, the mesh for the bolts was developed
manually. First, a node in the center was created. Next, the element of the bolt was
created by connecting the node in the center with the nodes of the main body which
interface with the bolts.
o Model the end-effector platform
The finite element model of the end-effector platform should be able to accurately
receive the transferred deformations from the bolts and the main body. First, the
circular platform was modeled through the solid modeling facility. Next, the elements
of circular platform were developed by use of the automatic meshing facility. At this
step, the nodes of the circular platform will in particular follow the location of the
nodes of the elements for the bolts which were previously defined.
Consequently, there were two sets of nodes developed. One set of nodes belonged to
the end-effector platform, while another set of nodes belonged to the elements of the
bolts (see Fig. 3.16). Finally, CP command was used to couple the end-effector
platform and the bolts-main body components in ANSYS.
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Prestress force
Prestress force
Prestress force
Prestress force
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62
Figure 3.14 The modeled PZT, plate, and compliant piece.
Figure 3.15 Modeling the boundary conditions and the bolts.
A
B
C
E
F
G
Nodes sharing between the elements of the bolt and the elements of the piece of material
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Figure 3.16 Modeling the end-effector platform.
Coupling two sets of nodes between the end-effector platform and the bolts-piece of material by use of CP command.
End-effector platform
Main body
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3.6 I llustrations
The purpose of this section is to illustrate the deformations of the RRR mechanism in
ANSYS for several critical positions. Generally, to achieve such positions, one needs
to apply the electrical voltages into one, or two, or three PZT actuators. Specifically,
such positions are divided into three categories: (1) the RRR mechanism positions
when only single PZT actuator is activated, as illustrated in Fig. 3.17a, Fig.3.17b and
Fig.3.17c, respectively; (2) the positions of the RRR mechanism when two PZT
actuators are activated. Fig. 3.18a, Fig 3.18b and Fig. 3.18c respectively illustrate the
RRR mechanism positions when PZT 1 and PZT 3, PZT 1 and PZT 2, and PZT 2 and
PZT 3 are activated ; (3) the RRR mechanism positions when all the PZT actuators
are activated, as illustrated in Fig. 3.19. The shape of the RRR mechanism prior to
the loading (so-called the original shape) is indicated by the discrete lines. In other
words, the parts of the RRR mechanism that do not situate within the discrete lines
have some deformation. In addition, the deformation shapes of the RRR mechanism
(see Figs. 3.17, 3.18 and 3.19) are presented by showing the RRR mechanism both
with and without the end-effector platform, for the purpose of clarity. The code for
this illustration is documented in Appendix C.
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Figure 3.17 The deformation of the RRR mechanism by activating the
single PZT actuator (a: PZT 1; b: PZT 2; c: PZT 3).
(a)
(b)
(c)
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Figure 3.18 The deformation of the RRR mechanism by activating two PZT
actuators (a: PZT 1 and 3; b: PZT 1 and 2; c: PZT 2 and 3).
(a)
(b)
(c)
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3.7 Summary and Discussion
The ANSYS finite element model of a compliant mechanism driven by three PZT
actuators (PZT RRR mechanism for short) was described in this chapter. This model
can be used for motion analysis without consideration of inertia. When the voltages
of the PZT actuators are prescribed, one can obtain the displacement at the end-
effector. From the literature review, it is believed that the model is unique for the
problem under study.
Traditionally, the motion analysis of the compliant mechanism is based on a concept
called the pseudo rigid body (PRB). In the PRB concept, a compliant mechanism is
first modeled by a PRB mechanism, and then motion analysis for the rigid body
mechanism is applied to the PRB mechanism (which is now a rigid body
mechanism). This procedure is not very accurate, as opposed to finite element
approach in general. In the finite element approach, the method developed by Zettl
[2003] can be considered as improvement of the PRB method, but it requires the
availability of more accurate 3D motion information which is usually obtained
Figure 3.19 The deformation of the RRR mechanism by activating
all PZT actuators.
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through 3D finite element analysis. 3D finite element modeling and simulation is
costly, which may not be available in practical design exercises.
The major disadvantage of a full finite element model, as the one presented in this
chapter, is of high computation resource as opposed to the PRB method. This has
restricted its application in real time control problem. Another limitation with the
model presented here is that it does not consider the inertia in the model.
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CHAPTER 4
NATURAL FREQUENCY AND STIFFNESS
4.1 Introduction
As reviewed in Chapter 2, the natural frequency and global stiffness of a compliant
mechanism are two very important design indices concerning dynamic behaviors of
the mechanism. These two indices are, indeed, traditional elements in structural
analysis, but they are not well-studied in the application of mechanisms (especially
compliant mechanisms). In this chapter, two approaches based on ANSYS are
presented. In particular Section 4.2 addresses the natural frequency, and Section 4.3
addressed the stiffness. Section 4.4 gives a summary.
4.2 Natural Frequency of Compliant Mechanisms
4.2.1 Basic concepts
ANSYS uses modal analysis to compute the natural frequencies of the RRR
mechanism. Modal analysis aims to find a set of parameters that represents the
vibration behavior of a structure. The set of parameters includes the natural
frequencies and mode shapes (patterns of vibration). In ANSYS, the modal analysis
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uses several numerical methods, which are Reduced Method, Subspace Method,
Block Lanczos Method, Damped Method, and QR Damped Method [ANSYS, 2004].
These methods are briefly discussed below.
Reduced Method
Guyan [1965] provides the theoretical basis of the Reduced Method in ANSYS. The
reduced method is basically a way to reduce the size of the matrices of a model for
the purpose of performing fewer computations. One distinctive feature of this method
is the implementation of so-called master degrees of freedom. The master degrees of
freedom are the key degree-of-freedoms that characterize the dynamic portion of the
model. Thus, instead of considering the entire finite element model, this method
requires a user to select the dynamic portion of the model. The key assumption in this
method is that the inertia forces on the so-called slave degrees of freedom (those
DOF being reduced out, thus the opposite of master degrees of freedom) are
negligible compared to elastic forces transmitted by the master DOF. Therefore, the
total mass of the structure is divided among only the master DOF. The net result is
that the reduced stiffness matrix is exact and the reduced mass matrix is approximate.
Consequently, the determination of the master degrees of freedom contributes
significantly in the accuracy that can be achieved with this method.
Subspace Method
Bath [1982] and Wilson et al. [1983] provide the theoretical basis of the Subspace
Method in ANSYS. The Subspace Method uses the subspace iteration technique,
which internally uses the generalized Jacobian iteration algorithm. The algorithm
seeks to solve the eigenvalue problem by use of full [K] and [M] matrices. The
Subspace Method is much slower than the Reduced Method. This method is typically
used in cases where high accuracy is required or where selecting master DOF is not
practical. However, this method is not applicable to the system that contains
piezoelectricity degrees of freedom.
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Block Lanczos Method
Rajakumar et al. [1991] and Grimes et al. [1994] provide the theoretical basis of
Block Lanczos method in ANSYS. To solve the eigenvalue problem, the Block uses
a combination of the automated shift strategy and the sturm sequence check strategy.
The two strategies aim to reduce the number of iterations in solving the eigenvalue
problem yet maintaining good accuracy.
Unsymmetric Method
This method uses a combination of the works of Rajakumar [1991] and Wilkinson
[1988]. The Unsymmetric Method, which also uses the full [K] and [M] matrices, is
meant for problems where the stiffness and mass matrices are unsymmetrical (for
example, acoustic fluid-structure interaction problems involving element FLUID 30
and MATRIX 27).
Damped Method
The works of Rajakumar and Ali [1992] and Wilkinson [1988] provide the basis for
the damped method. This method is applicable to problems where the damping is
considered. The method considers full matrices [K], [M], and [C].
QR Damped Method
The QR damped method combines the advantages of the Block Lanczos Method and
the Hessenberg Method. The Hessenberg Method can be found in [Kardestuncer et
al. 1987]. The main idea of the QR damped method is to approximately represent the
first few complex damped eigenvalues by a linear combination of a small number of
eigenvectors of the corresponding undamped system.
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Power Dynamics Method
The power dynamics method uses a combination of the subspace method and the
preconditioned conjugate gradient (PCG). The PCG is basically an iterative solver in
which the equations are not solved directly but instead, an initial estimate of the
solution is made, and a computational procedure is defined whereby the estimate is
improved until it satisfies the equations within some specified tolerance. The power
dynamic method is considerably faster than the subspace method and the block
lanczos in computation speed, because this method does not perform a Sturm
sequence check and uses a reduced mass matrix (instead of a full mass matrix).
Accuracy achieved with this method may be compromised because of the reduced
mass matrix.
The Block Lanczos method was chosen in this work to compute the natural
frequency. It is noted that for a compliant mechanism, each set of prescribed
actuations corresponds to a “ frozen” configuration. The natural frequency is then
associated with this configuration. In other words, the modal analysis as described
before will be applied on this structure. Fig. 4.1 is a flow chart is to compute the
natural frequency of the “ frozen” structure. Each step in the flow chart is explained
below.
4.2.2 Procedure
(1) Finite element modeling
A finite element model of the compliant mechanism must be available prior to
computing the natural frequency of the compliant mechanism. For the PZT RRR
mechanism, the finite element model presented in Chapter 3 was used.
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(2) Selection of a calculation method
As discussed previously, there are seven calculation methods in ANSYS. The Block
Lanczos method was selected to compute the natural frequency of the RRR
mechanism. This was based on the following reasons. The Reduced Method and the
Power Dynamic Method were not chosen because of the accuracy concern. Due to
the complex nature of the RRR mechanism, the process of locating master degrees of
freedom on the RRR mechanism was difficult. The Power Dynamic Method was not
chosen because it also uses a reduced mass approximation, instead of a full mass
matrix. The damped method and the QR damped method were not chosen because
they are not designed for obtaining the natural frequency information. The
unsymmetric method was not chosen because none of the components of RRR
mechanism that has the unsymmetric stiffness. Finally, the Subspace Method was not
chosen because the RRR mechanism entails the piezoelectricity degree-of-freedom.
(3) Activation of the prestress option is activated
The prestress option is an option in ANSYS to calculate the natural frequencies of
this system. The prestress option in ANSYS considers the possibility that a system is
prestressed prior to computing the natural frequencies of the system. The prestress
option needs to be activated in ANSYS due to the fact that in the modal analysis the
system is assumed to be stress-free (by default). However, the PZT RRR mechanism
is pre-stressed to become a “ frozen” structure. Therefore for the RRR mechanism (or
in general compliant mechanism), the prestress option should be considered.
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4.2.3 Validation
The work performed by Kitis and Lindenberg [1989] was used to validate the
proposed approach described in Section 4.2.2. They used the transfer matrix method
to compute the natural frequencies of a four-bar mechanism (see Fig. 4.2) for several
configurations. Different configurations are determined by giving different values of
the crank angle (θ2). A finite element model of this mechanism can be found from
Appendix E.6. In their work, they used two modeling strategies for a component. The
first, two segments were chosen to model each component (also called model 2); the
second, three segments were chosen to model each component (also called model 3).
Our finite element model corresponded to their model 2 (two finite elements used for
one component) and model 3 (three finite elements used for one component). Fig.
4.3, Fig. 4.4, and Fig. 4.5 present the results of comparison between their approach
and our approach, in particular the first mode, the second mode, and the third mode,
respectively. Fig 4.3 shows a strong correlation between their approach and our
Figure 4.1 The procedure to compute the system frequency.
(1) Finite Element Modeling
(2) Selection of the calculation method
(3) Activation of the prestress option
(4) Process of the modal analysis
Prescription of a “ frozen configuration”
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approach when the crank angle is from 0 to 250 degrees. However, two of the largest
deviations occur when the crank angles are 300 degrees (8 rad/sec or 1.273 Hz) and
350 degrees (30 rad/sec or 4.77 Hz), respectively. Fig 4.4 indicates also some good
agreement. For model 2 in Fig. 4.4a, the results of natural frequency for the second
mode show that the largest deviation occur when the crank angle is 100 degrees (9
rad/sec or 1.432 Hz), while for model 3 in Fig. 4.4.b, the results of the natural
frequency for the second mode show that the largest deviation occur when the crank
angle is also 100 degrees (6.74 rad/sec or 1.073 Hz).
Figure 4.2 A four-bar mechanism.
Coupler
Crank Follower
Pin Joint 1
Pin Joint 2
θ2 θ4
θ3
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(a)
(b)
Figure 4.3 The result of comparison for the first mode.
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Fig 4.5 indicates the least agreement compared to the results presented in Fig. 4.3 and
Fig. 4.4. For model 2 in Fig. 4.5a, the results of the natural frequency for the third
mode show that the largest deviation occur when the crank angle is 250 degrees
(235.14 rad/sec or 37.43 Hz), while for model 3 in Fig. 4.5.b, the results of natural
frequency for the third mode show that the largest deviation occur when the crank
angle is also 250 degrees (238.18 rad/sec or 37.91 Hz).
(a)
(b)
Figure 4.4 The result of comparison for the second mode.
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.
In summary, the two approaches have obtained some good agreement for the first
two modes. For the third mode, the results have shown less agreement. The
disagreement of the two approaches increases when the mode of shape increases,
which is evidenced by the results shown in Fig. 4.5. However, the two approaches
obtained the similar minimum and maximum values, for the third mode
(approximately at 380 rad/sec to 710 rad/sec). Also, there is a trend that is when the
number of elements (or segments) increases, the two approaches agree more. The
(a)
(b)
Figure 4.5 The result of comparison for the third mode.
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finite element model is generally more accurate than the transfer matrix model
because the latter introduces more assumptions with regard to ideal status of a
structure system. Our approach is thus reliable for predicting the natural frequency of
a compliant mechanism.
4.2.4 Results
The computation of the RRR mechanism has been performed for several critical
conditions. Fig. 4.6, Fig. 4.7, and Fig. 4.8 present the natural frequencies of the RRR
mechanism as only single PZT is activated: PZT 1, PZT 2, and PZT 3, respectively.
Fig. 4.9, Fig. 4.10, Fig. 4.11 show the natural frequencies of the RRR mechanism as
two PZT actuators are activated: PZT 1 and PZT 2, PZT 1 and PZT 3 , and PZT 2
and PZT 3, respectively. Fig. 4.12 presents the natural frequencies of the RRR
mechanism as three PZT actuators are activated: PZT 1, PZT 2 and PZT 3.
In summary, in the current design of the RRR mechanism, the natural frequencies of
the first and second modes are relatively independent of the configurations of the
system, and they are also very close (~268 Hz). While the natural frequency of the
third mode is relatively dependent on the configuration of the system; specifically
ranging from 402 Hz to 405 Hz depending on different configurations.
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Figure 4.6 The natural frequencies of the PZT 1 actuation.
Figure 4.7 The natural frequencies of the PZT 2 actuation.
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Figure 4.8 The natural frequencies of the PZT 3 actuation.
Figure 4.9 The natural frequencies of the PZT 1 and 2 actuation.
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Figure 4.10 The natural frequencies of the PZT 1 and 3 actuation.
Figure 4.11 The natural frequencies of the PZT 2 and 3 actuation.
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4.3 System Stiffness
4.3.1 Basic concepts
Elsewhere in Chapter 2 (Section 2.6) the concept of system stiffness has been
elaborated. It is understood that the interest of stiffness in mechanisms or robots lies
in the so-called global stiffness at the end-effector. In the following, a procedure
based on a general-purpose finite element program (i.e., ANSYS) is proposed.
4.3.2 Procedure
A planar mechanism is considered without loss of generality.
Figure 4.12 The natural frequencies of the PZT 1, 2, and 3 actuation.
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Step 1:
Add FX (F1) = 1, FY (F2) = 1, and M (F3) =1, respectively, on the end-effector.
Step 2:
Execute the finite element model with F1, F2, and F3, and get the end-effector
displacement (position and orientation): X ( 1x ), Y ( 2x ), θ ( 3x ), respectively, i.e.,
=
13
12
11
1
x
x
x
Xr
;
=
23
22
21
2
x
x
x
Xr
;
=
33
32
31
3
x
x
x
Xr
In the above ijx denotes the displacement i produced due to the force j. There should
be the following equation:
=−=∆
3
2
1
0 ][)(
F
F
F
CXXXrrr
(4.1)
=
332313
322212
312111
][
xxx
xxx
xxx
C (4.2)
[K] = [C]-1 (4.3)
In the above, [K] is the global stiffness matrix, and 0Xr
is the displacement of the
end-effector at a particular configuration.
Step 3:
Find the Jacobian matrix for the mechanism system;. The PRBM of the RRR
mechanism was developed by Zou [2000]; see the schematic diagram of this
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mechanism in Fig. 4.13. In the case of the PZT RRR mechanism, Jacobian matrix,
0lJ can be found as follows [Zou, 2000]:
0lJ = -
0
AB
R
Lsin( 3ψ )
−−−
−−−
)sin(3
1
)sin(3
1
)sin(3
1
)sin(3
1)cos(
3
3)sin(
3
2)sin(
3
1)cos(
3
3
)cos(3
1)sin(
3
3)cos(
3
2)cos(
3
1)sin(
3
3
ψψψ
ψψψψψ
ψψψψψ
RRR
(4.4)
where ABL : the length of the link ii BA , i=1,2,3,
BCL : the length of the link iiCB , i=1,2,3,
R : the length of iOC , i=1,2,3,
ψ : 21 ψψ + , in which 1ψ and 2ψ are illustrated in Fig. 4.10,
and 3ψ : )sin
arcsin( 22
AB
BC
L
L ψψ ×+ .
Step 4:
Find the system global stiffness limits; that is, first get the eigenvalues from the
matrix [K][T
lJ 0 ][ 0lJ ]; second, get minγ (the minimum eigenvalue) and maxγ (the
maximum eigenvalue).
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4.3.3 Validation
The work of Sanger et al. [2000] was chosen to validate our approach. The
mechanism has two degrees-of-freedom: q1 and q2; while P denotes the position of
the end-effector in the global coordinate system (X-O-Y). The length of links OQ,
QP and PO are 10 cm, 10 cm, and 14.14 cm, respectively; while the stiffness of the
actuators (q1 and q2 ) are 10 N/cm.
Four methods were employed to find the global stiffness for this mechanism.
C2
C1
A2
A3
2ψ
1ψ
X
Y
1 θ
2 θ
O 3 θ
A1
B1
B3 C3
B2
Figure 4.13 The PRBM of the RRR mechanism [Zou, 2000].
Figure 4.14 A two-legged planar manipulator.
P (x,y)
q1 q2
α1 α2
O Q X
Y
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(1) The procedure proposed in [Sanger et al., 2000]
Sanger et al. [2000] derived a formula to compute the stiffness matrix at the end-
effector as follows:
K = JkJT – ηλT (4.5)
where K : the stiffness matrix at the end-effector,
J : the Jacobian matrix relating to the actuated joints,
JT : the transpose of the Jacobian matrix relating to the
actuated joints,
η : the matrix describing the incremental change in J due to
changes in the unactuated joint displacements,
and λT : the transpose of the the Jacobian matrix relating to the
unactuated joints.
At the position of P (10,10) cm, the stiffness matrix is equal to
200
020N/cm.
When P = (18,25) cm , the stiffness matrix is
7076.195290.0
5290.08813.18 N/cm.
(2) Our approach
When P = (10, 10) cm, K =
155
55 N/cm. When P = (10, 10) cm, K =
652.15648.7
648.7348.4 N/cm.
(3) The procedure proposed in [Dawe, 1984]
The matrix displacement approach was proposed in [Dawe, 1984]. The following
steps were taken in this procedure. First, each link of the two-legged planar
manipulator (see Fig. 4.14) is put into the following table.
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Table 4.1 Table of structural elements as P (10, 10) cm
Element The stiffness
Of actuator (k)
The angle of
link
The element stiffness
k0
OP 10 N/cm 45 degrees 10
5.05.0
5.05.0N/cm
PQ 10 N/cm 90 degrees 10
10
00N/cm
The presented element stiffness in Table 4.1 is based on:
k0 = k ×
ααα
ααα2
2
sinsincos
sincoscos (4.6)
Next, the global stiffness matrix is assembled as follows.
Because the two-legged planar manipulator is constrained in O and Q (see Fig. 4.10),
the rows and columns corresponding to O and Q can be eliminated. Therefore, the
global stiffness matrix of the two-legged planar manipulator is [ PQQP kk 00 + ], or equal
to
K =
155
55 N/cm.
A similar procedure can be applied for P (18, 25) cm, which results in
O P Q
O
P
Q
++
+
QOPQQPQO
PQPQQPPO
PQOPOPQO
kkkk
kkkk
kkkk
0000
0000
0000
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K =
652.15648.7
648.7348.4 N/cm.
(4) The procedure proposed in [Gosselin, 1990]
Gosselin [1990] derived an equation for the planar manipulators, as presented below.
K = k JTJ (4.7)
Note that Eqn. (4.7) is part of Eqn (4.5). Eqn (4.7) was also presented in [Sanger et
al., 2000] for the application of the two-legged planar manipulator application.
JkJT =
++++
222
112
222111
222111222
112
sinsincossincossin
cossincossincoscos
kkkk
kkkk
αααααααααααα
(4.8)
The use of Eqn. (4.11) results also in the similar stiffness matrices that were obtained
with the second and the third approaches.
From the above comparison, the first approach does not produce the same result as
the other three. Our approach to compute the global stiffness matrix for the compliant
mechanism agrees with the third and fourth approaches and is thus reliable. Further,
our approach may be better than the third and fourth approaches because they have
introduced some assumptions of a mechanism under investigation. For example,
Gosselin [1990]’s approach assumed that all actuators should have the same axial
stiffness along with their actuating axes and the stiffness of the link and other passive
joints are not considered. The approach proposed by Gosselin and Zhang [1999]
extended the one by Gosselin [1990] by considering the stiffness of the link
component. It should be noted that both the third and the fourth approaches are
strongly associated with the structures that contain truss members or beam members.
So, these approaches are inherently not suitable for the compliant mechanism which
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is often not characterized by beam members or truss members. Gosselin and Zhang
[1999] mentioned in their work that the problem with the first approach lies in the
consideration of the unactuated joint.
4.3.4 Results
The minimum and maximum stiffness of the PZT RRR mechanism for different
configurations was calculated using our approach, and their results are shown in Fig.
4.15, Fig. 4.16, Fig. 4.17, Fig. 4.18, Fig. 4.19, Fig. 4.20, and Fig. 4.21.
Figure 4.15The stiffness of the PZT 1 actuation.
Minimum stiffness
Maximum stiffness
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Figure 4.16The stiffness of the PZT 2 actuation.
Figure 4.17 The stiffness of the PZT 3 actuation.
Maximum stiffness
Minimum stiffness
Maximum stiffness
Minimum stiffness
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Figure 4.18 The stiffness of the PZT 1 and 2 actuation.
Figure 4.19 The stiffness of the PZT 2 and 3 actuation.
Minimum stiffness
Maximum stiffness
Maximum stiffness
Minimum stiffness
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Figure 4.20 The stiffness of the PZT 1 and 3 actuation.
Figure 4.21 The stiffness of the PZT 1, 2, and 3 actuation.
Maximum stiffness
Minimum stiffness
Maximum stiffness
Minimum stiffness
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4.4 Conclusion
New methods for predicting the natural frequency and the global stiffness of a
compliant mechanism have been developed, respectively. They are easily
implemented on a general purpose finite element program, such as ANSYS. These
two methods have been validated by comparing the simulation results produced by
them with the known reference results. It should be noted that the popular paradigm
for analysis of compliant mechanism, called pseudo rigid body model, is generally
not suitable to the calculation of the natural frequency and global stiffness for
compliant mechanisms.
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CHAPTER 5
EXPERIMENTAL VALIDATION
5.1 Introduction
In this chapter, measurement results based on a prototype PZT RRR mechanism is
presented. This is essential in order to validate the theoretical developments
described in the previous chapters. Measurements were performed at both the
actuator level and the end-effector level. Section 5.2 presents the test bed which
includes the instrument and related fixture devices. Section 5.3 presents the
measurement results together with the simulation results based on the models (both
the other previous model and the model developed with this thesis study. A
discussion about these results is also included in Section 5.3. Section 5.4 is a
conclusion.
5.2 Measurement Test-bed Set-up
5.2.1 Measurement at the end-effector
Fig. 5.1 illustrates the measurement instrumentation system to measure the end-
effector motion. The manufacturer of this system is KAMAN and its model name is
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SMU 9000-15N-001. The system is based on the magnetic-induction principle,
specifically eddy current phenomenon. An eddy (swirl) current is a local electric
current induced in a conductive material by the magnetic field produced by the active
coil (see Fig. 5.2). When an AC current flows in a coil in certain proximity to a
conductive material, the developed magnetic field in the coil will induce circulating
(eddy) currents in the conductive material. The electromagnetic sensors sense
impedance variation as the gap changes and then the calibration box translates the
impedance variation into a usable displacement signal. The measurement resolution
of the system is 0.1 microns. The system is hereafter also called the induction sensor.
There are some requirements that need to be met regarding the mounting of the
induction senor with respect to an object to be measured (target for short). They are
(1) distance requirement, (2) sensor mounting requirement, (3) parallelism
requirement, (4) target requirements, and (5) sensor to sensor proximity requirement.
These requirements are briefly discussed below.
Figure 5.1 SMU 9000-15N-001 [Kaman, 2000].
Calibration box
Sensor
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Figure 5.2 Eddy current behavior [Kaman, 2000].
The distance requirement refers to the minimum and maximum distances within
which the target and the sensor must stay (see Fig. 5.3). The sensor mounting
requirement is about the minimum empty area around the tip of the sensor (see Fig.
5.4). The parallelism requirement is about the maximum allowable tilt angle of the
target (i.e., 3 degrees, see Fig. 5.5). The target requirements are restrictions on (a) the
material of the target, (b) the minimum diameter of the target, and (c) the thickness of
the target. The material requirement for the target is Aluminum T6. Fig. 5.6
illustrates the diameter of the target and the thickness of target. Fig. 5.7 shows the
sensor to sensor proximity requirement.
Figure 5.3 Required distance between sensor and target [Kaman, 2000].
! " #
$
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Figure 5.4 Sensor mounting requirement [Kaman, 2000].
Figure 5.5 Parallelism requirement [Kaman, 2000].
Figure 5.6 Target requirements [Kaman, 2000].
# ∅∅∅∅
∅∅∅∅
$
Sensor
T a rg et
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Figure 5.7 Requirement for sensor to sensor proximity [Kaman, 2000].
KAMAN [2000] defines calibration as a means to verify that system output (in the
form of an output voltage) relates to a known physical displacement with a known
degree of accuracy. KAMAN provides the calibration equation as follows:
Y= 2.191e-5 X5-4.259e-4 X4 +3.497e-3 X3 – 1.297e-X2
+ 8.674e-2 X- 4.92 e-4 (5.1)
where X : the resulted sensors’ reading in forms of the voltages,
and Y : the computed distance
Note that Eqn. (5.1) cannot be directly applied to the RRR mechanism because its
environment and that of the manufacturer are different. This type of difference is
sensitive to the accuracy of measurement. There were two options for coping with
this problem. One was to make the application measurement environment similar to
the manufacturer environment, which is costly. The other was to recalibrate the
measuring instrument. The latter option was chosen in this study because an X-Y-Z
stage with 0.2 µm displacement resolution (manufacturer: Newport company; model:
M-461) is available to this study, which can be used to act as a reference
measurement. The recalibration was conducted by following the steps shown in Fig.
5.8.
∅∅∅∅
∅∅∅∅
Sensor 1
Sensor 2
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Table 5.1 presents the calibration result. The detailed physical setting for the
recalibration can be found in Appendix D.
Figure 5.8 The procedure to calibrate SMU 9000-15N-001 for the
RRR mechanism application.
1. Varying the distance between sensor and target within the distance requirement of induction sensor by use of the X-Y-Z motion stage
2a. Record the voltage results of induction sensor
3a. Use Eqn. (5.1) to compute the displacements
2b. Record the increment displacement results from the X-Y-Z motion stage
3b. Compute the displacements
4. Error = item 3a – item 3b
5. Find the smallest error.
Induction sensor X-Y-Z motion stage
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Configuration PZT 1 (volts) PZT 2 (volts) PZT 3 (volts)
1 0 0 0
2 16 16 16
3 32 32 32
4 48 48 48
5 64 64 64
6 80 80 80
7 96 96 96
8 112 112 112
9 128 128 128
In order to measure three motions (X, Y, θ) simultaneously, three induction sensors
are required, see Fig. 5.9. From Fig. 5.9 one can get the end-effector motion as
follows:
Figure 5.9 The RRR mechanism set-up in experiment.
Sensor 1
Workbench
Sensor 3
Y axis of end-effector
X axis of end-effector
B
C
A O
∆X1 ∆X2
∆X3
L Sensor 2
Table 5.1 The configurations of the RRR mechanism
when all the PZT actuators are activated.
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where ∆X and ∆Y are displacements along the X-axis and the Y-axis, respectively,
while ∆ γ is the angular displacements of the end-effector. All these displacements
are with respect to the X-Y coordinate system in the measurements situation.
To physically realize the above scheme for obtaining the end-effector motion, a
workbench was designed; see Fig. 5.10.
Figure 5.10 Adjustable Workbench.
∆X = 2
32 XX ∆+∆ (5.5)
∆Y = 122 )( XLAO ∆+∆+ γ (5.6)
∆γ = )(tan 321
BC
XX ∆−∆− −
(5.7)
Sensors BASE
DOVETAILS
TARGETS BLOCK
Sensor caps
Mechanical clamps
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The workbench has a certain degree of adaptability in the sense that it can
accommodate different physical configurations of a target. The detailed design of the
workbench can be found in Appendix C.
The reference coordinate system for the measurement was different from that for the
simulation based on the model developed in Chapter 3. Such a difference is shown in
Fig. 5.11. Conversion of the simulation result from the reference for the simulation to
the reference for the measurement was conducted. It is noted that the X axis in the
experiment perpendicularly cuts the half of the line BC. The conversion equation is
given as below:
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X-axis in ANSYS
Y-axis in ANSYS X-axis in experiment
Y-axis in experiment
B
C
A
(a)
Sensor 1
Workbench
Sensor 3
Y-axis in experiment
X-axis in experiment
B
C
A O
∆X1 ∆X2
∆X3
Sensor 2
X-axis in ANSYS Y-axis in ANSYS
Figure 5.11 The reference in simulation versus the reference in measurement (a: the
reference in the measurement; b: the reference in the simulation).
(b)
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5.2.2 Measurement at the actuator level
The measurement at the actuator level concerns the displacement of the PZT actuator
and was directly obtained from the strain gauge on the PZT actuator. Fig. 5.12 shows
a schematic diagram of the measurement at the actuator level. The manufacturer of
the strain gauge is Vishay Measurements Group, Inc, the model name is EA-06-
125TG-350, and its accuracy is ± 0.05 %.
Figure 5.12 A schematic diagram of the measurement system [Handley et al. 2002].
5.3 Results and Discussions
The measurement on a prototype PZT RRR mechanism was currently limited to the
configurations where all the three PZT actuators are activated in the same amount
(see Table 5.1 for a list of these configurations). This is because a poor repeatability
has been found in other configurations, which is further attributed to the fatigue
problem with any compliant mechanism. Specifically, the measurement at the end-
effector level was made prior to over thousand times of operations for the
measurement at the actuator level.
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The reliability test on these configurations was ensured by rotating the placement of
the RRR mechanism. The first measurement was made at the original placement,
while the second measurement was made by rotating the mechanism, say A to B, B to
C, and C to A (see Fig. 5.13). The results from these two replacements were shown in
Fig. 5.14. The average errors for the two replacements are about 1 micron for X-
direction, 0.2 micron for Y-direction and 0.1 mrad for rotational direction. The errors
are very small, which confirms the high repeatability of the measurements at these
configurations.Fig. 5.15 presents the comparison of the results at the end-effector, at
which both the simulations (Zou [2000], Zettl [2003], and the model developed in
this thesis) and the measurement are shown. From the figure it can be seen that the
results of this thesis are the closest to the measurement data, but still some significant
deviations (the maximum deviation: 1.49 microns) exist. Such deviations are
explained as follows. In the experiment, the RRR mechanism has some non-identical
pre-load forces acted on the PZT actuators and the unmeasured initial deformation of
the RRR mechanism due to the limitations of the manufacturing tool to produce the
identical thickness of the thin plates. This is because that the piezoelectric actuators
were assembled / pressed into the RRR mechanism by hand (to create a tight fit), in
order to ensure the PZT actuators stay in its individual slots within the main body.
B
C
A O X-axis of end-effector
Y-axis of end-effector
Figure 5.13 Rotating the positions of A, B and C.
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However, the information regarding the accurate values of the thickness of the thin
plates and the initial deformation of the RRR mechanism are necessary in finite
element modeling, which is in fact either not very accurate (due to the limitations of
the measurement instrument to measure the thickness of the thin plates) or
unavailable (due to the unavailability of a measurement instrument to measure the
initial deformation experienced by the RRR mechanism due to prestress). This also
explains the deviation of the results among Zou [2000], Zettl [2003], the
measurement and the present model. The results presented in the present model are
closer to the measurement results because the model has, to a certain extent, captured
the non-identical thickness of the thin plates; whereas Zou [2000] and Zettl [2003]
did not capture the information regarding the assembly of the RRR mechanism. This
means that their results for the deformations of the RRR mechanism in X and Y
directions (as all the PZT actuators are activated on the same input loadings as shown
in Fig. 5.15) are insignificant compared to the measurement and the present model.
Fig. 5.16 presents the comparison of measurement and simulation at the actuator
level. In particular, the axial deformations of the piezoelectric actuators within the
PZT-RRR mechanism as only a single piezoelectric actuator was activated (PZT 3)
are presented. Similar to the results presented at the end-effector level, the results
presented in the present model lie closer to the measurement results. To conclude, the
simulation results validate the ‘uncoupling’ nature of the inactivated piezoelectric
actuators that were observed during the experiment [Handley et al., 2002]. This result
should enhance the reliability of the model developed in this thesis.
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Figure 5.14 Check the data repeatability of the end-effector deformations
in experiment.
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Figure 5.15 Comparison of the end-effector deformation results.
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Figure 5.16 The comparison of measurement and simulation at the actuator level
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5.4 Summary and Conclusion
The experiment test bed was established and described in this chapter, which
included the measurement both at the actuator and the end-effector levels. The three
simulation models, the model by Zou [2000], the model by Zettl [2003], and the one
developed in this thesis study, were compared with the real measurements for two
purposes. The first purpose is to explore the accuracy of the models, and the second
purpose is to explore the design and assembly of the compliant mechanism. The
comparison has shown that our model corresponds strongest with measurement. This
is because our model has captured the physical property of the PZT actuator more
fully, which includes (1) the pre-stress behaviour of the PZT actuator, and (2) the
physical property of the piezoelectric material. Furthermore, the comparison also
reveals the importance of design and assembly of the PZT actuator with the
compliant mechanism. The current practice with the PZT RRR mechanism produces
considerable uncertainty in achieving non-uniformity among three PZT actuators.
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CHAPTER 6
CONCLUSIONS AND FUTURE WORK
6.1 Overview
This thesis presents a study toward a finite element approach to compliant
mechanisms. In the current literature, the compliant mechanism is usually analyzed
by use of the so-called pseudo rigid body (PRB) method. The basic idea of the PRB
method is to lump continuous materials into a set of lumped materials that are
connected by the rigid members. The problems with the PRB method are
uncontrolled inaccuracy and high computation resource due to a complex dynamic
model. Furthermore, the PRB method appears to be too complex to calculate the
natural frequency and the global stiffness of the compliant mechanism.
A pioneer study using a general-purpose finite element program for analysis of
compliant mechanisms was conducted at the Advanced Engineering Design
Laboratory (AEDL) at the Department of Mechanical Engineering at the University
of Saskatchewan [Zou et al., 2000]. A recent study at AEDL refers to Ref. [Zettl,
2003]. These studies have not considered the PZT actuator in a systematic way.
Furthermore, there has been no published method, to the best of the author’s
knowledge, that computes the natural frequency and the global stiffness of a
compliant mechanism using a finite element approach. Based on a detailed analysis
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of literature presented in Chapter 2, the following objectives were proposed for the
research presented in this thesis.
Objective 1: To develop a more accurate finite element model of the compliant
mechanism for motion analysis with special attention to capturing the physical
behaviour of the piezoelectric actuator, which is embedded in and drives the
compliant mechanism.
Objective 2: To develop a more reliable test bed for the compliant mechanism with
the objective to provide a test environment for the validation of the model for motion
analysis.
Objective 3: To develop methods based on finite element analysis for predicting the
global stiffness and natural frequency properties of the compliant mechanism.
These objectives have been achieved. The following are the details.
A literature review (Chapter 2) was conducted to confirm the statement of the
objectives. Specifically, it was found that the finite element model of the compliant
mechanism, incorporating the PZT actuator, was not previously reported in literature.
The method for the calculation of the natural frequency and the global stiffness of the
compliant mechanism using any general purpose finite element method was not
reported elsewhere.
In Chapter 3, a finite element model for the compliant mechanism with consideration
of the PZT actuator was presented. The model was implemented in the ANSYS
environment; in particular, a type of element which deals with interdisciplinary
domains (mechanical, electrical, etc), available in ANSYS, was employed. The pre-
stress in the PZT actuator was also considered.
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In Chapter 4, a finite element approach for calculating the natural frequency of the
compliant mechanism was presented. The key to this problem is to view a
mechanism as a series of “ frozen” configurations, at each of which the mechanism
becomes a structure. There are several solvers in ANSYS for calculating the natural
frequency problem for a structure. They were reviewed and analyzed, resulting in the
employment of the Block Lanczos Method. Another piece of the study presented in
Chapter 4 is the calculation of the global stiffness of the compliant mechanism. A
novel procedure based on the finite element method was formulated. The matrix size
to calculate the global stiffness of the compliant mechanism is at most six by six.
In Chapter 5, a test bed was established for validating the displacement analysis of
the compliant mechanism with the developed finite element approach described in
Chapter 3. The comparison also included the studies conducted by Zou [2000] and
Zettl [2003], respectively. The result of comparison shows that the method developed
in this thesis has improved the other existing methods in terms of agreement between
the measured result and the simulation result.
The finite element approach developed in this thesis was implemented by the
general-purpose finite element program system ANSYS. The study presented in this
thesis concludes:
(1) The finite element model for the PZT driven compliant mechanism should
consider the piezoelectric material property more fully. The use of a block
element or a spring element to simulate the PZT actuator stiffness does not work
very well.
(2) The pre-stress in the piezoelectric element has significant influences over the
accuracy of the finite element model.
(3) The piezoelectric element driven RRR mechanism has a kinematical uncoupling
property among three actuators.
(4) The design of the assembly of the piezoelectric element and the compliant
mechanism needs to be revisited carefully. The current design can lead to
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considerable uncertainty in maintaining the symmetry of the whole mechanism
(with respect to the center of the material).
6.2 Contr ibutions
The main contributions of this thesis are described below:
(1) A more accurate model of the piezoelectric element driven compliant mechanism
based on a special type of element available in ANSYS that represents the
properties that are related to two different disciplines was developed.
(2) A finite element approach to compute the global stiffness of the compliant
mechanism, which captures the stiffness of both the actuator and the compliant
material was developed.
(3) A test bed upon which comparison of the models and the real measurements can
be made was developed.
6.3 Future Work
The optimal design of the RRR mechanism warrants investigation. One of the design
objectives is to have a large range of micro-motion without compromising the
accuracy of motion. However, the large range may very likely involve reduced
system stiffness. Therefore, a design trade-off is highly needed. The optimal design is
to get the best trade-off. Furthermore, the optimal design may also be integrated with
the optimal planning of motion to meet the requirement on the end-effector regarding
motion and force.
The uncoupling property among actuators needs to be investigated together with the
topology and geometry of the mechanism. A further verification of whether this
property is exclusively related to the symmetry of the compliant mechanism system.
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The current design has not considered the inertia term in the context of a general
dynamic model of a mechanism system )(][][][ tfxkxcxm =++ &&& . The finite element
model needs to be extended to consider the inertia term for motion analysis of the
compliant mechanism.
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Technology.
Wilson, E. L. and Itoh, Tetsuji, 1983. "An Eigensolution Strategy for Large Systems”,
Computers and Structures, 16 (1-4), pp. 259-265.
Zettl, B., 2003. Effective Finite Element Modeling of Micro-Positioning Systems.
M.Sc., University of Saskatchewan, Saskatoon, Canada.
Zou, J. 2000. Kinematics, Dynamics, and Control of a Particular Micro-Motion
System. M.Sc., University of Saskatchewan, Saskatoon, Canada.
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Appendix A: Physical Design of the RRR Mechanism
The RRR mechanism studied in this thesis consists of the following components:
(1) Main body;
(2) Three PZT actuators;
(3) One end effector platform;
(4) Three bolts; and
(5) Three thin metal plates.
They are assembled into a mechanism, as shown in Fig. A.1
Figure A.1 The assembly of the RRR mechanism.
Bolt
Bolt Bolt
PZT actuators
End-effector platform
Main body
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A.1 The main body
Fig. A.2 illustrates the topology and geometry of the main body, where holes A, B
and C are used to fix the main body onto a ground, while holes 1, 2 and 3 are used to
fix the main body to the end-effector platform. As such, when the main body deforms
under the action of the PZT actuator, the end-effector will exercise motion with
respect to the ground. This main body was made up of bronze material and named C
61000 based on the UNS (Unified Numbering Standard). The main body has the
following properties.
• 8% Aluminum Bronze,
• Modulus Elasticity: 117 x 109 Pa, and
• Density: 7.78 x 103 Kg/m3.
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1
2
3
A
B C
TOP VIEW
SOLID VIEW
10.000
Figure A.2 The main body (all the units are mm)
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A.2 The PZT actuator
The PZT actuator used in this thesis study is the product of Tokin manufacturer
[Tokin, 2000], specifically the model AE0505D16. Both the geometry and material
properties of this actuator were given in Chapter 2. The displacement of the PZT
actuator is measured by a strain gage. In this study the strain gage is a product of
Vishay Measurements Group, Inc.; specifically the model EA-06-125TG-350. Prior
to measurement of the PZT actuator by use of the strain gage, one should pay close
attention to the process of attaching the strain gage onto the PZT actuator due to the
fact that the strain gage has capability of measuring the smallest effects of an
imperfect bond. The imperfect bond will result in the inaccurate reading results. The
procedure of gluing the strain gage onto the PZT actuator can be classified into
surface preparation, strain gage bonding, and soldering. In surface preparation, the
objectives are to develop chemically clean surface (by use of special chemical agent,
M-Bond 200, to remove the oil and the grease), to create the appropriate surface
roughness (by use of a special abrading paper, silicon-carbide paper, to remove rust
and paint), to build the correct PH of the surface (by use of the special agent, M-Prep
Neutralizer 5A, to neutralize the PH of the surface) of the object that will be
measured (i.e., the PZT actuator) and to create the clear and visible gage layout lines
for positioning the strain gage onto the PZT actuator. In strain gage bonding, it is
important to ensure that the bonded strain gage stays still on the surface (visible gage
layout lines in particular) that is going to be measured due to the fact that its
performance is absolutely dependent on the bond between itself and the test part.
Thus, the procedure that was discussed in [Vishay Measurements Group, 1992]
should be followed. The purpose of soldering is to install the wires into the glued
strain gage such that the specified resistance requirement from the manufacturer is
met. There is a measurement instrument recommended by manufacturer (Model 1300
Gage Installation Tester) to ensure if the specified resistance requirement from the
manufacturer is met (10,000 to 20,000 ohms).
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Fig. A.3 shows the installed strain gage on the PZT actuator. The strain gages are
glued to two sides of the PZT actuator and act as a pair. Each pair is wired to their
own strain gage conditioner. The strain gage conditioner is to compute the response
of the PZT actuators by use of the calibration equations. These calibration equations
are then used to determine the displacement of each PZT. Based on the experiment,
the strain gage has the position resolution of 10 nm.
Figure A.3 The attached strain gage on the PZT actuator.
A.3 The end effector platform
The topology and geometry of the end-effector is given in Fig. A.4. It is noted that
the holes 1, 2, 3 on the end-effector platform are assembled with the corresponding
holes on the base. The material properties of the end-effector platform are given as
follows:
• Standard name based on ASM (American Society for Metals Specialty
Handbook): Aluminum 6061-T6,
• Modulus Elasticity: 69 x 109 Pa,
• Proportional limit: ( pσ )≤ 275 x 106 Pa, and
• Density : 2.768 x 104 Kg/m3 .
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Figure A.4 The end-effector platform.
A.4 The bolt
Fig. A.5 illustrates the geometric information of a bolt. The three identical bolts are
to attach the end-effector platform and the main body at holes 1, 2 and 3,
respectively. The material properties presented as follows:
• Standard name based on ASTM (American Society for Testing and Materials:
10/32 Fine Thread Screw,
• Modulus elasticity: 358.28 x 106 Pa, and
• Density: 8.780 x 103 kg/m3 .
3D VIEW
SOLID VIEW
2
3
1
TOP VIEW
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Figure A.5 Geometric boundaries of a bolt.
A.5 The adjusting (thin metal) plate and the pre-stress
The thin metal plate is a plate inserted between the PZT actuator and the main body
to create fitting tolerance as illustrated in Fig. A.6. The rationales of employing such
components are available in Section 3.4.1. The material properties of the thin metal
plate are illustrated as follows:
Figure A.6 Location of the thin metal plate within the RRR.
• Standard name based on ASTM (American Society for Testing and Materials:
C1018, and
• Modulus elasticity: 344.5 x 106 Pa.
SOLID VIEW 3D VIEW
Thin metal plate
PZT
A main body
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Torsional loads are not an issue because the driving elements (piezoelectric actuators)
for the RRR compliant mechanism only exhibit the axial loading. Eliminating shear
loading was done in the assembly process of the RRR compliant mechanism. During
the assembly process, the compliant mechanism was attached using a special
manufactured plate (a simple structural plate that has uniform thickness along its
length inserted between PZT and the part of compliant piece) that was manufactured
such that the axis polarization of piezoelectric actuator aligns with the center axis of
flexure hinge. After that, the prestress state was applied on compliant mechanism.
Fig. A.7 illustrates the compliant piece under prestress state.
The prestress state was realized by inserting a thin metal plate into the gaps between
actuators and actuators’ slots within compliant piece, to create tight fit (causing
actuators to be compressed). When the compliant piece is in motion, the prestress is
to ensure the position and orientation of the actuators (that have been aligned with the
bracket) do not change with respect to the center axis of flexure hinge (in order to
Figure A.7 Compliant piece under prestress state.
Actuator
Compliant piece
Thin metal plate
Centre axis of flexure hinge
Uniform thickness structural plate.
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prevent shear loads from occurring on flexure hinges’ surfaces), to prevent separation
between actuators and compliant piece, and to prevent tension condition in actuators
from occurring that can damage actuators. The measurement of the pre-deformation
due to the plate is given in Appendix B.
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Appendix B: The measurement of the pre-deformation
Step1. Obtained the original length of the PZT slot (see Fig. B.1)
The length of the slots of the PZT actuator prior to pre-deformation was obtained
through the original design drawing of the RRR mechanism. The length of each PZT
slot prior to pre-deformation is 21 mm.
Step 2. Measured each slot of the pre-deformed RRR mechanism
By use of a measurement instrumentation (a digital caliper), the length of each slot of
the pre-deformed RRR mechanism was obtained. The lengths of the pre- deformed
slots of PZT 1, PZT 2, and PZT 3 are 20.24 mm, 20.51 mm, and 19.97 mm
respectively.
Figure B.1 The slots of the PZT actuators.
Slot 1
Slot 2
Slot 3
Slot 1
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Step 3. Calculated the pre-deformed forces
Based on the following equation
F = E × A × l
l∆ From Eqn. (3.20)
where F : the prestress force or load (N),
E : the Young modulus of the PZT material (4.4e10 N/m2),
A : the cross sectional area of the PZT actuator (25e-6 m2),
l : the length of the PZT slot (21e-3 m),
and l∆ : the displaced length of the PZT slot due to the prestress (m).
The values of l∆ were obtained by subtracting the original length of the PZT slot (as
discussed in step 1) with the pre-deformed length of the PZT slot (as discussed in
step 2. The values of ∆l1, ∆l2, and ∆l3 are 0.76e-3 m, 0.49e-3 m, and 1.03e-3 m,
Figure B.2 The pre-deformed slot of the PZT actuators.
Pre-deformed slot 1
PZT1
PZT2
PZT3
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respectively. Finally, the values of the pre-deformed forces of PZT 1, PZT 2, and
PZT 3 can be calculated. The pre-deformed forces of PZT 1, PZT 2, and PZT 3 are
3.981e4 N, 2.56667e4 N, and 5.395 e4 N, respectively.
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Appendix C: Design of a Workbench
There are the following requirements for assembly of the sensor and the target, i.e.,
(1) Distance requirement
(2) Sensor mounting requirement
(3) Parallelism requirement
(4) Target requirements
(5) Sensor to sensor proximity requirement
The detailed design of an adjustable workbench upon which the sensor and the target
can be assembled in meeting the above requirements is described below.
C1. Distance requirement
Figure C.1 Required distance between sensor and target [Kaman, 2000].
Fig. C.1 shows the required distance between the sensor and the target. There are two
types of required distance; the offset distance and the working range distance. The
offset distance is a minimum distance that needs to be maintained between the sensor
and the target without degrading the reading accuracy, while the working range
distance refers to the range limits (the minimum and maximum limits) that have to be
maintained between the sensor and the target during the measurement. So, the
distance between the sensor and target need to be situated from 0.25 mm to 1.25 mm.
Sensor
O f f set ( 0 . 2 5 m m )
Work i ng R a ng e ( 0 -1 m m )
T a rg et
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Figure C.2 Distance between sensor and target in the workbench
As illustrated in Fig C.2, the distance between the dovetail and the target is 9.75 mm.
This value is not only to overcome the distance requirement, but also to provide some
sufficient space for the adjustment of the sensor’s position on the dovetail for the
calibration purpose.
C2. Sensor mounting requirement
Figure C.3 Sensor mounting requirement [Kaman, 2000].
Fig. C.3 shows the minimum empty area, which is represented by variables W and L,
around the tip of the sensor. This area needs to be maintained. The dovetail was
designed to maintain L and W.
9.75 mm
Sensor
1 . 5 t o 2 X ’ s Sensor c oi l ∅∅∅∅
Sensor
C ond u c t i v e M ou nt 2 . 5 t o 3 X ’ s Sensor c oi l ∅∅∅∅
W
L
Target
Dovetail
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Figure C.4 Geometric boundaries of dovetail and sensor.
Figs. C.4a and C.4b present the geometric boundaries of the dovetail and those of the
sensor. Variable D (see Fig. C.4a) is an important variable of the dovetail’ s variables
in maintaining variable L (see Fig. C.3). In this thesis work, the variable D is
manufactured to be 12.840 mm. The dovetail supports the part of sensor body in the
value of 12.840 mm; consequently, a part of the sensor body that is not supported by
the dovetail is about 17.67 mm. This value is sufficient not only to compensate L
Variables Values (mm)
A 41.579
B 67.539
C 10
D 12.840
E 10
F 10
G φ 5
H 600
I φ 5
J 58.879
(a)
(b)
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(calculated to be 8.26 mm due to the fact the sensor’s coil diameter is 4.13 mm), but
also to provide some adequate adjustment for the purpose of sensor calibration.
Figure C.5 The height of the sensor.
Fig. C.5 is to determine the height of the sensor that is measured from the block. This
height is manufactured to be 28.912 mm in order to compensate half of W (calculated
to be 12.39 mm divided by two, which is equal to 6.195 mm). Note that W is divided
by two because W applies to the both areas of interest, in particular the area below
the sensor’s position and the area above the sensor’s position. The height of the
sensor that is measured from the block (28.912 mm) should compensate half of the W
(6.195 mm).
C3. Parallelism requirement
Figure C.6 Parallelism requirement [Kaman, 2000].
To maintain the parallelism requirement, a guiding plate is inserted between the
sensor and the block prior to fixing the sensor onto the dovetail. During the
measurement, this plate is removed.
Sensor
T a rg et
3 d eg rees t a rg et t i l t
Height = 28.912 mm
Block
Base
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Figure C.7 Solution to compensate the parallelism requirement.
C4. Target requirement
The target requirements comprises of the material of the target, the minimum
diameter of the target, and the thickness. As previously mentioned, Kaman
determines that the material requirement for the target is Aluminum T6, while the
minimum diameter of the target and the thickness requirement is re-illustrated in Fig.
C.8.
Figure C.8 Target requirements [Kaman, 2000].
Block
Base
Plate
! " #∅∅∅∅
Sensor $# " "&% #
T a rg et Thickness = 0.4572 mm
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Var iables A B φ C φ D E F G H
Values
(mm)
45 19.05 5 8.5 15 17 9.525 9.525
Figure C.9 Geometric boundaries of a target.
Fig. C.9 presents the geometric boundaries of a target. The minimum thickness of a
target is 0.4572 mm (see Fig. C.8). To compensate the geometric boundaries of a
target, this value is in particular variable B is 19.05 mm. As for the minimum
diameter size of the target (12.39 mm based on the computation), such a requirement
has been overcome in the target by providing the variables B and F with the values of
19.05 mm and 17 mm respectively.
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C5. Sensor to sensor proximity requirement
It is important to maintain the distance between the sensors in order to prevent the
electromagnetic interference between sensors that leads to the reading accuracy
degradation. Fig. C.10 shows the sensor to sensor proximity requirements. Based
on the calculation from Fig. C.10, the minimum distance between two sensors is
12.39 mm. From Fig. C.11, the length of each dovetail is 41.579 mm, while the space
that separates the dovetails is 5 mm. Thus, the distance between the sensors in the
workbench is equal to 46.579 mm, which is sufficient to compensate the requirement
for sensor to sensor proximity, which is 12.39 mm.
Sensor 1
Sensor 2
∅∅∅∅
∅∅∅∅
Figure C.10 Requirement for sensor to sensor proximity [Kaman, 2000].
41.579 mm 41.579 mm
Figure C.11 The distance between two sensors.
Distance between two sensors
5 mm
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Figure C.12 An assembled workbench.
The assembly diagram of the workbench is shown in Fig. C 12. Components are,
shown in Fig. C.13, while details can be found in the following files
(“workbench1.doc” and “workbench2.doc”) in the attached CD disk.
BASE
DOVETAILS
TARGETS
BLOCK
Magnify the view for the above circled part
Magnify the view for the above circled part
3-RRR
Sensors
Sensor caps
Mechanical clamps
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Figure C.13 Components of a workbench.
(a) (b) (c)
(d)
(e)
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The dovetail (see Fig. C.13a) functions to hold sensor in its place. The dovetail has
three taps and two holes on the top. The taps are to accommodate the screws’
entrance, in particular to fix the dovetail into the base. Two taps were used to fix the
sensor cap on the top. The sensor cap (see Fig. C.13b) has three taps. Two taps in the
corners are to accommodate the screws’ entrance, while the tap in the middle is to
facilitate the screw’s entrance in order to facilitate fixing the sensor so that the sensor
remains on its place. Note that the screws are made of rubber material in order the
fixing process do not harm the sensor’s body. The target (see Fig. C.13c) is placed on
the end-effector platform of the RRR mechanism (see Fig. C.12). The base (see Fig.
C.13d) is a foundation of all components. The base has two slots. By use of the
screws, these slots are to facilitate the dovetails adjustment prior to fixing the
dovetail onto the base. Block (see Fig. C.13e) is a platform as a place for attaching
the 3-RRR. The block is the last component to be manufactured, due to the fact that
the block must not only provide the proper fixing of the RRR mechanism (with three
special designed taps) but also to ensure that the height of the sensors is aligned
with the height of the targets (measured from the base).
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Appendix D: Calibration of the sensor
This calibration has the following procedure: (1) Varying the distance between the sensor and the target within the distance
requirement of the sensor (SMU 9000-15N-001) by use of the X-Y-Z stage (M-461,
Newport Company)
It is noted that the main reason that this X-Y-Z stage was used is that this stage has
0.2 microns, compared to the RRR mechanism (1 micrometer accuracy).
In this step, a target is placed onto the platform of the X-Y-Z stage, while a sensor is
placed onto the platform of the microscope (see Fig. D.2). Next, the distance between
the target and the sensor is adjusted by actuating the left-manipulator.
Figure D.1 Find the accurate distances between sensors and targets.
Sensor 1
Block
Sensor 2
Sensor 3
Y axis of end-effector
X axis of end-effector
B
C
A O
Target
Target
Target
Accurate initial distance?
Accurate initial distance?
Accurate initial distance?
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(2) Record the reading results
Both the reading results from the sensor and the X-Y-Z stage were recorded
simultaneously. Note that the sensors displayed the values in the form of voltages,
while the X-Y-Z stage presented the values in the form of microns.
(3) Calculate errors (results deviation)
For the purpose of calculating errors, the obtained results from the sensor are then
converted by use of Eqn. (5.1), prior to subtracting them from the reading results of
the X-Y-Z stage.
(4) Locate the smallest errors
From the experiment, it was obtained that the smallest errors for sensors 1, 2 and 3
are presented in the below table.
Figure D.2 The X-Y-Z stage (M-461, Newport Company).
Left-hand manipulator
Platform of left-hand manipulator Platform of the microscope
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Table D.1 The smallest errors of the three sensors.
One should pursue the sensor output readings relating to the smallest errors prior to
perform the measurement by tapping carefully the position of the sensor on the
dovetail.
Fig. D.2 presents the flow chart of the process of measuring the end-effector
displacements to facilitate general understanding of the process. Every performed
measurement consists of two processes, which are calibration process and
measurement process.
Sensors Sensor output readings Smallest errors
Sensor 1 4.791 volts 0.0787 microns
Sensor 2 5.181 volts 0.0494 microns
Sensor 3 5.47 volts 0.0154 microns
Figure D.3 The measuring process of the end-effector displacements
PZT 1 PZT 2 PZT 3
Sensor 1 Sensor 2 Sensor 3
Step 1
Step 2
∆X1 ∆X2
∆X3
X Y Rotation
End-effector
Step 4
Step 5
YA YB
YC
Step 3
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During the calibration process, the distance between the sensor and its corresponding
target should be conditioned such that the sensor output readings in Table D.1 are
achieved. Note that the PZT actuators should not have any loading during the
calibration process. After the sensor output readings in Table D.1 are achieved, Eqn
(5.1) is to convert those readings into the computed distances (represented as YA, YB,
and Yc in Fig. D3). YA, YB, and Yc are 332.851, 362.009, and 384.249 microns,
respectively. These values are the initial positions of the targets. Consequently, ∆X1,
∆X2 and ∆X3 that represent the deformations of the three targets are zero. The end-
effector deformations, which can be computed by use of Eqns. (5.5), (5.6), and (5.7),
are zero as well.
During the measurement process, the PZT 1, PZT 2 and PZT 3 would have different
type of loadings. This will affect the sensor output voltage readings (Sensor 1, Sensor
2, and Sensor 3, as shown in Fig. D.3). This will result in the new values of YA, YB,
and Yc. The differences between the new values of YA, YB, and Yc and the initial
positions of the targets (332.851, 362.009, and 384.249 microns) result in the non-
zero values of the deformations of the targets (∆X1, ∆X2 and ∆X3). Accordingly, the
end-effector deformations can also be computed.
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Appendix E: ANSYS codes
E.1. Introduction
This Appendix contains the ANSYS codes that will generate the finite element
models in Chapter 3 and Chapter 4. This Appendix comprises of:
1. PIEZMAT macro; the codes supplied by the ANSYS technical support.
2. Three dimensional model of the PZT actuator.
3. Two dimensional model of the PZT actuator.
4. The PZT-RRR mechanism model.
5. The modified MATLAB codes based on the original [Zou, 2000] to be used
along with the PZT-RRR mechanism model; for computing the system
stiffness
6. The case study of the four-bar mechanism model; for validating the procedure
to compute the natural frequency in Chapter 4.
7. The case study of the two-bar mechanism model; for validating the procedure
to compute the system stiffness in Chapter 4.
E.1. PIEZMAT macro
!MACRO TO CREATE PIEZOELECTRIC INPUT FROM !MANUFACTURER'S DATA PROCESSING WILL REQUIRED THE !INVERSION OF THE MANUFACTURER'S COMPLIANCE MATRIX INTO !ANSYS STIFFNESS FORM ! ! 5/25/99 - Initial Release ! 2/14/00 - Revision to remove 5.5 inversion technique ! Add arg4, arg5 and arg6 to convert units after processing ! User must supply conversion factors ! arg4 to convert stiffness matrix ! arg5 to convert piezoelectric matrix ! arg6 to convert permittivity matrix ! Add Fatal if negative permittivity ! THE FOLLOWING MATRICES WILL BE NEEDED ! MPIEZC - MANUFACTURER'S COMPLIANCE MATRIX - 6 X 6 ! MPIEZD - MANUFACTURER'S PIEZOELECTRIC MATRIX - 6 X 3
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! MPIEZDT - TRANSPOSE OF MPIEZD - 3 X 6 ! MPIEZP - MANUFACTURER'S DIELECTRIC MATRIX - 3 X 3 ! PIEZCINV - ANSYS STIFFNESS MATRIX - 6 X 6 ! PIEZEANS - ANSYS PIEZOELECTRIC MATRIX - 6 X 3 ! PIEZEPP - ANSYS DIELECTRIC MATRIX - 3 X 3 ! ! DIFFERENT PROCESSING FOR BATCH AND INTERACTIVE ! !!!!/nopr *GET,IBATCH,ACTIVE,,INT ! ! START BY SAVING CURRENT PARAMETERS TO A FILE ! PARSAV,ALL,PIEZTEMP,PAR ! * IF,IBATCH,LE,.5,THEN PARRES,CHANGE,ARG2,ARG3 PIEZMAT=ARG1 *ELSE ! ! ASK IF DATA IS ON FILE OR TO BE ENTERED ! !!!! CREATION OF INTERACT MACRO LET AS AN EXERCISE !!!! *ASK,IFILE,ENTER FILE NAME, 0 FOR KEYBOARD ENRTY,'0' !!!! * IF,IFILE,EQ,'0',THEN !!!! *ASK,PIEZMAT,ENTER THE MAT NUM FOR PIEZO DATA,1 ! READ DATA FROM KEYBOARD !!!! *ELSE ! PARRES,CHANGE,IFILE,PAR PARRES,CHANGE,ARG2,ARG3 PIEZMAT=ARG1 !!!! *ENDIF *ENDIF ! AT THIS POINT THE MATRICES HAVE BEEN DEFINED ! INVERT THE C MATRIX ! *GET,REVN,ACTIVE,,REV * IF,REVN,GE,5.6,THEN ! *MOPER,PIEZCINV(1,1),MPIEZC(1,1),INVERT ! ! FOR 5.5, INVERT BY APDL ! *ELSE *MSG,ERROR
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CONTACT ANSYS TECHNICAL SUPPORT FOR A 5.5 VERSION OF THIS MACRO *ENDIF ! FORM THE ANSYS PIEZOELECTRIC MATRIX ! PIEZEANS = PIEZCINV * MPIEZD ! *MOPER,PIEZEANS(1,1),PIEZCINV(1,1),MULT,MPIEZD(1,1) ! ! FORM THE DIELECTRIC MATRIX ! FIRST TRANSPOSE MPIEZD, ! THEN POST MULTIPLY TO PIEZEANS ! FINALLY SUBTRACT FROM MPIEZP ! *MFUN,MPIEZDT(1,1),TRAN,MPIEZD(1,1) *MOPER,PIEZEPP(1,1),MPIEZDT(1,1),MULT,PIEZEANS(1,1) *DO,II,1,3 *VOPER,PIEZEPP(1,II),MPIEZP(1,II),SUB,PIEZEPP(1,II) *ENDDO ! ! STOP IF PERMITTIVITY IS NEGATIVE *VWRITE ( PERMITTIVITY MATRIX) *VWRITE,PIEZEPP(1,1), PIEZEPP(1,2), PIEZEPP(1,3) (3E14.7) JMTPIEZ=0 JMTPIEZ=MIN(JMTPIEZ,PIEZEPP(1,1)) JMTPIEZ=MIN(JMTPIEZ,PIEZEPP(2,2)) JMTPIEZ=MIN(JMTPIEZ,PIEZEPP(3,3)) * IF,JMTPIEZ,LT,0,THEN *MSG,ERROR,JMTPIEZ PERMITTIVITY VALUE = %E IS LESS THAN ZERO *ENDIF ! CONVERT UNITS IF APPROPRIATE ! PIEZCINV FROM N/SQ M TO LBF/SQ IN ! PIEZEANS FROM COULOMBS/SQ M TO COULOMBS/SQ IN ! PIEZEPP FROM FARADS/M TO FARADS/IN * IF,ARG4,GT,0,THEN *DO,II,1,6 *VOPER,PIEZCINV(1,II),PIEZCINV(1,II),MULT,arg4 *ENDDO *DO,II,1,3 *VOPER,PIEZEANS(1,II),PIEZEANS(1,II),MULT,arg5 *VOPER,PIEZEPP(1,II),PIEZEPP(1,II),MULT,arg6 *ENDDO *ENDIF
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! ! CREATE THE TBDATA COMMANDS ! TB,PIEZ,PIEZMAT ! TBDATA,1,PIEZEANS(1,1),PIEZEANS(1,2),PIEZEANS(1,3) TBDATA,4,PIEZEANS(2,1),PIEZEANS(2,2),PIEZEANS(2,3) TBDATA,7,PIEZEANS(3,1),PIEZEANS(3,2),PIEZEANS(3,3) TBDATA,10,PIEZEANS(4,1),PIEZEANS(4,2),PIEZEANS(4,3) TBDATA,13,PIEZEANS(5,1),PIEZEANS(5,2),PIEZEANS(5,3) TBDATA,16,PIEZEANS(6,1),PIEZEANS(6,2),PIEZEANS(6,3) ! MP,PERX,PIEZMAT,PIEZEPP(1,1) MP,PERY,PIEZMAT,PIEZEPP(2,2) MP,PERZ,PIEZMAT,PIEZEPP(3,3) ! TB,ANEL,PIEZMAT TBDATA,1,PIEZCINV(1,1),PIEZCINV(2,1),PIEZCINV(3,1),PIEZCINV(4,1),PIEZCINV(5,1),PIEZCINV(6,1) TBDATA,7,PIEZCINV(2,2),PIEZCINV(3,2),PIEZCINV(4,2),PIEZCINV(5,2),PIEZCINV(6,2) TBDATA,12,PIEZCINV(3,3),PIEZCINV(3,4),PIEZCINV(3,5),PIEZCINV(3,6) TBDATA,16,PIEZCINV(4,4),PIEZCINV(4,5),PIEZCINV(4,6) TBDATA,19,PIEZCINV(5,5),PIEZCINV(5,6) TBDATA,21,PIEZCINV(6,6) ! ! CLEAN UP AFTER PIEZOELECTRIC MATERIAL DEFINITION ! SAVE AND RESUME TO RESTORE PREVIOUS PARAMETERS ! PARSAV,ALL,PIEZANS,PAR PIEZMAT= MPIEZC(1,1)= MPIEZD(1,1)= MPIEZP(1,1)= MPIEZDT(1,1)= PIEZCINV(1,1)= PIEZEANS(1,1)= PIEZEPP(1,1)= PARRES,CHANGE,PIEZTEMP,PAR /GOPR !
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E.2. Three dimensional model of the PZT actuator
! For ANSYS educational version 8.1, only electrical properties (dielectric !constant) require conversion (by use of PIEZMAT macro) !130=number of layers of the PZT TOKIN AE0505D16 !1.4 = the manufacturer’s constant /PREP7 PZT=0 !Adjust voltage here !!PZT MATERIAL PROPERTIES!!! TB,PIEZ,1,,,1 TBMODIF,1,1, TBMODIF,1,2, TBMODIF,1,3,(-287e-12) TBMODIF,2,1, TBMODIF,2,2, TBMODIF,2,3, TBMODIF,3,1, TBMODIF,3,2, TBMODIF,3,3,130*(635e-12)*1.4 TBMODIF,4,1, TBMODIF,4,2, TBMODIF,4,3, TBMODIF,5,1,(930e-12) TBMODIF,5,2, TBMODIF,5,3, TBMODIF,6,1, TBMODIF,6,2, TBMODIF,6,3, ! !MECHANICAL PROPERTIES TB,ANEL,1,1,21,1 TBTEMP,0 TBDATA,,(14.8e-12) ,,,,, TBDATA,,,,,,,(18.1e-12)*130*1.4 TBDATA,,,,,,, TBDATA,,,,,,, MPTEMP,,,,,,,, MPTEMP,1,0 ! !ELECTRICAL PROPERTIES MPDATA,PERX,1,,2536.045198 ! This has been converted by use of PZT macro MPDATA,PERY,1,,2536.045198 ! MPDATA,PERZ,1,,130*1.4*2815.141243 !
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ET,1,SOLID5 ! 3-D COUPLED-FIELD SOLID, PIEZO OPTION KEYOPT,1,1,3 !!!!Modeling PZT ! Actual measurement is 5 x 5 x 20 mm k,1,0,0,0 k,2,5e-3,0,0 k,3,5e-3,5e-3,0 k,4,0,5e-3,0 k,5,0,0,20e-3 k,6,5e-3,0,20e-3 k,7,5e-3,5e-3,20e-3 k,8,0,5e-3,20e-3 type,1 mat,1 real,1 NKPT,NODE,ALL ! CREATE NODES ON KEYPOINTS E,1,2,3,4,5,6,7,8 ! CREATE ELEMENT !Apply voltages FINISH /SOL FLST,2,4,1,ORDE,2 FITEM,2,5 FITEM,2,-8 /GO D,P51X,VOLT,0 FLST,2,4,1,ORDE,2 FITEM,2,1 FITEM,2,-4 /GO D,P51X,VOLT,PZT
E.3. Two dimensional model of the PZT actuator
/PREP7 VOLTAGES=40 length=1 width=1 ET,1,PLANE13 KEYOPT,1,1,7 ! UX, UY, XOLT KEYOPT,1,3,0 ! PLANE STRAIN !KEYOPT,1,3,2 ! PLANE STRESS
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/PREP7 TB,PIEZ,1,,,1 TBMODIF,1,1,130*1.35*(635e-12) TBMODIF,1,2, TBMODIF,1,3, TBMODIF,2,1,-287e-12 TBMODIF,2,2, TBMODIF,2,3, TBMODIF,3,1, TBMODIF,3,2, TBMODIF,3,3, TBMODIF,4,1, TBMODIF,4,2,930e-12 TBMODIF,4,3, TBMODIF,5,1, TBMODIF,5,2, TBMODIF,5,3, TBMODIF,6,1, TBMODIF,6,2, TBMODIF,6,3, TB,ANEL,1,1,21,1 TBTEMP,0 TBDATA,,130*1.4*18.1e-12 TBDATA,,14.8e-12 ,,,,, TBDATA,,,,,,, TBDATA,,,,,,, MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,PERX,1,,130*1.4*2815.045198 MPDATA,PERY,1,,2536.045198 MPDATA,PERZ,1,, type,1 mat,1 real,1 !CREATE KEYPOINTS! K,1,0,0 K,2,20e-3,0 K,3,20e-3,5e-3
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K,4,0,5e-3 !CREATE LINES! LSTR, 1, 2 LSTR, 2, 3 LSTR, 3, 4 LSTR, 4, 1 !!!! dividing length line !!!!!! FLST,5,2,4,ORDE,2 FITEM,5,1 FITEM,5,3 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,length, , , , ,1 !!!! dividing width line !!!!!! FLST,5,2,4,ORDE,2 FITEM,5,2 FITEM,5,4 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,width, , , , ,1 !!!create area!!! FLST,2,4,4 FITEM,2,1 FITEM,2,3 FITEM,2,4 FITEM,2,2 AL,P51X !!!MESHING AREA!! MSHKEY,0 CM,_Y,AREA ASEL, , , , 1 CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1
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CMDELE,_Y2 FINISH !!!!!apply PZT volts /SOLU FLST,2,2,1,ORDE,2 FITEM,2,1 FITEM,2,4 /GO D,P51X,VOLT,VOLTAGES FLST,2,2,1,ORDE,2 FITEM,2,2 FITEM,2,-3 /GO D,P51X,VOLT,
E.4. The PZT-RRR mechanism model
PZT1=0 PZT2=100 PZT3=100 long=1 short=1 !* **Creating geometric boundaries of compliant mechanism***************** /PREP7 /UNITS,SI ! SPECIFY MKS UNITS *SET,R,32e-3 ! *SET,pi,3.14159 !constant for converting deg. to rad *set,l,9e-3 ! *SET,rr,1e-3 ! *SET,h,10e-3 ! *SET,t,0.8e-3 ! *SET,g,(h-t-2*rr)/2 ! *Set,w,8e-3 ! *SET,g2,(w-t-2*rr)/2 ! *SET,lab,17e-3 ! *SET,lbc,11e-3 ! *SET,h1,l+rr+h/2+lab ! Declaration of parameters *SET,h2,R-w-lbc ! *SET,flg,3*rr+t/2 ! *SET,r0,4.5e-3 ! *SET,clr,0
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k,1,0,0 ! k,2,l*cos(-30/180*pi),l*sin(-30*pi/180) ! k,3,l*cos(-30/180*pi)+1e-3*cos(60/180*pi),l*sin(-30*pi/180)+1e-3*sin(60*pi/180) ! k,4,r*cos(-30/180*pi)+1e-3*cos(60/180*pi),r*sin(-30*pi/180)+1e-3*sin(60*pi/180) ! k,5,R,0 ! Key k,6,R,l ! point k,7,r-g2,l ! assign k,8,r-g2-rr,l+rr ! ment k,9,r-g2,l+2*rr ! k,10,R,l+2*rr ! k,11,R,l+rr+lab+h/2 ! k,12,R-w,h1 ! k,13,R-w,h1-g ! k,14,R-w-rr,h1-g-rr ! k,15,R-w-2*rr,h1-g ! k,16,R-w-2*rr,h1 ! k,17,h2,h1 ! k,18,h2,h1-g ! k,19,h2-rr,h1-g-rr ! k,20,h2-2*rr,h1-g ! k,21,h2-2*rr,h1 ! k,22,h2-h-2*rr,h1 ! k,23,h2-h-2*rr,h1-h-2*1e-3 ! k,24,h2-2*rr,h1-h-2*1e-3 ! k,25,h2-2*rr,h1-h+g ! k,26,h2-rr,h1-h+g+rr ! k,27,h2,h1-h+g ! k,28,h2,h1-h ! k,29,R-w-2*rr,h1-h ! k,30,r-w-2*rr,h1-h+g ! k,31,r-w-rr,h1-h+g+rr ! k,32,r-w,h1-h+g ! k,33,r-w,l+r0+t/2+rr+clr ! k,34,r-w-rr,l+r0+t/2+clr ! k,35,r-w-2*rr,l+r0+t/2+rr+clr ! k,36,r-w-2*rr,l+r0+2.5e-3+clr ! k,37,r-w-2*rr-2e-3,l+r0+2.5e-3+clr ! k,38,r-w-2*rr-2e-3,l+r0-2.5e-3+clr ! k,39,r-w-2*rr,l+r0-2.5e-3+clr ! k,40,r-w-2*rr,l+r0-t/2-rr+clr ! k,41,r-w-rr,l+r0-t/2+clr ! k,42,r-w,l+r0-t/2-rr+clr ! k,43,r-w,l+2*rr ! k,44,r-w+g2,l+2* rr ! k,45,r-w+g2+rr,l+rr !
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k,46,r-w+g2,l ! k,47,0,l ! k,48,r-12e-3,0 ! k,49,r-w-2*rr-lbc-h/2,l+rr+lab ! k,50,r-w-2*rr-lbc-h/2+3e-3,l+rr+lab-5e-3 ! k,51,r-w-2*rr-lbc-h/2-3e-3,l+rr+lab-5e-3 ! k,52,-1e-3,11e-3,, ! k,53,-1e-3,16e-3,, ! l,1,2 ! l,2,3 ! l,3,4 ! larc,4,5,1,32e-3 ! l,5,6 ! l,6,7 ! larc,7,8,6,rr ! Creating lines between key points larc,8,9,6,rr ! l,9,10 ! l,10,11 ! l,11,12 ! l,12,13 ! larc,13,14,16,rr ! larc,14,15,12,rr ! l,15,16 ! l,16,17 ! l,17,18 ! larc,18,19,21,rr ! larc,19,20,17,rr ! l,20,21 ! l,21,22 ! l,22,23 ! l,23,24 ! l,24,25 ! larc,25,26,28,rr ! larc,26,27,24,rr ! l,27,28 ! l,28,29 ! l,29,30 ! larc,30,31,29,rr ! larc,31,32,29,rr ! l,32,33 ! larc,33,34,14,rr ! larc,34,35,14,rr ! l,35,36 ! l,36,37 !
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l,37,38 ! l,38,39 ! l,39,40 ! larc,40,41,48,rr ! larc,41,42,48,rr ! l,42,43 ! l,43,44 ! larc,44,45,47,rr ! larc,45,46,47,rr ! l,46,47 ! l,47,1 ! lsel,all !Create area of 1/3 of compliant al,all !mechanism without holes aplot circle,48,2.5e-3 ! circle,49,2.5e-3 ! circle,50,1e-3 ! circle,51,1e-3 ! circle,1,2.5e-3 lplot !Generate areas representing holes al,48,49,50,51 ! al,52,53,54,55 ! al,56,57,58,59 ! al,60,61,62,63 ! al,64,65,66,67 ! CSYS,1 ! Copying one area FLST,3,7,5,ORDE,2 ! to create one FITEM,3,1 ! whole area FITEM,3,-7 ! AGEN,3,P51X, , , ,120, , ,0 ! FLST,2,3,5,ORDE,3 ! Subtracting holes in FITEM,2,1 ! compliant piece FITEM,2,7 ! FITEM,2,13 ! FLST,3,9,5,ORDE,9 ! FITEM,3,2 ! FITEM,3,4 ! FITEM,3,-5 ! FITEM,3,8 ! FITEM,3,10 !
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FITEM,3,-11 ! FITEM,3,14 ! FITEM,3,16 ! FITEM,3,-17 ! ASBA,P51X,P51X ! FLST,2,3,5,ORDE,2 FITEM,2,19 FITEM,2,-21 ASBA,P51X, 6 /REPLOT FLST,2,3,5,ORDE,3 FITEM,2,1 FITEM,2,-2 FITEM,2,4 ASBA,P51X, 12 /REPLOT FLST,2,3,5,ORDE,2 FITEM,2,5 FITEM,2,-7 ASBA,P51X, 18 /REPLOT FLST,2,3,5,ORDE,3 ! Selecting 3 areas FITEM,2,1 ! FITEM,2,-2 ! FITEM,2,4 ! AGLUE,P51X ! Gluing compliant mechanism CSYS,1 ! Change coordinate system FLST,3,1,3,ORDE,1 ! FITEM,3,49 ! KGEN,3,P51X, , , ,120, , ,0 ! Copy kps GPLOT FLST,2,3,5,ORDE,3 FITEM,2,1 FITEM,2,5 FITEM,2,-6 FLST,3,3,5,ORDE,3 FITEM,3,3 FITEM,3,9 FITEM,3,15 ASBA,P51X,P51X GPLOT
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!****************Creating KP for thin plate*** ******************* CSWPLA,11,0,1,1, K,210,19e-3,16e-3,, K,211,19e-3,11e-3,, CSWPLA,11,1,1,1, FLST,3,2,3,ORDE,2 FITEM,3,210 FITEM,3,-211 KGEN,3,P51X, , , ,120, , ,1 !!!* ***************** *************create area for PZT + thin plate CSWPLA,11,0,1,1, FLST,2,4,3 FITEM,2,53 FITEM,2,210 FITEM,2,211 FITEM,2,52 A,P51X FLST,2,4,3 FITEM,2,210 FITEM,2,37 FITEM,2,38 FITEM,2,211 A,P51X CSWPLA,11,1,1,1, FLST,3,2,5,ORDE,2 FITEM,3,1 FITEM,3,3 AGEN,3,P51X, , , ,120, , ,0 !* ***********glue PZT-Thinplate-compliant FLST,2,9,5,ORDE,2 FITEM,2,1 FITEM,2,-9 AGLUE,P51X !* **********create elements for PZT ET,1,PLANE13 KEYOPT,1,1,7 ! UX, UY, VOLT DOF KEYOPT,1,3,0 ! PLANE STRAIN ASSUMPTION /PREP7
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TB,PIEZ,1,,,1 TBMODIF,1,1,130*1.4* (635e-12) TBMODIF,1,2, TBMODIF,1,3, TBMODIF,2,1,-287e-12 TBMODIF,2,2, TBMODIF,2,3, TBMODIF,3,1, TBMODIF,3,2, TBMODIF,3,3, TBMODIF,4,1, TBMODIF,4,2,930e-12 TBMODIF,4,3, TBMODIF,5,1, TBMODIF,5,2, TBMODIF,5,3, TBMODIF,6,1, TBMODIF,6,2, TBMODIF,6,3, TB,ANEL,1,1,21,1 TBTEMP,0 TBDATA,,0.0013*130*1.4*18.1e-12 TBDATA,,14.8e-12 ,,,,, TBDATA,,,,,,, TBDATA,,,,,,, MPTEMP,,,,,,,, MPTEMP,1,0 MPDATA,PERX,1,,130*1.4*2815.045198 ! MPDATA,PERY,1,,2536.045198 MPDATA,PERZ,1,, !* ************************************mesh area PZT 1 FLST,5,2,4,ORDE,2 FITEM,5,69 FITEM,5,134 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,long, , , , ,1 !!!!!long=5 divisions along longer body of PZT FLST,5,2,4,ORDE,2 FITEM,5,114 FITEM,5,136
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CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,short, , , , ,1 !!!!!short=3 divisions along longer body of PZT MSHKEY,0 CM,_Y,AREA ASEL, , , , 1 CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !!!!!!!mesh area PZT 2 FLST,5,2,4,ORDE,2 FITEM,5,212 FITEM,5,214 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,long, , , , ,1 FLST,5,2,4,ORDE,2 FITEM,5,213 FITEM,5,215 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,short, , , , ,1 MSHKEY,0 CM,_Y,AREA ASEL, , , , 8 CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2
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!!!!!!!mesh area PZT 3 FLST,5,2,4,ORDE,2 FITEM,5,200 FITEM,5,205 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,long, , , , ,1 FLST,5,2,4,ORDE,2 FITEM,5,201 FITEM,5,208 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,short, , , , ,1 MSHKEY,0 CM,_Y,AREA ASEL, , , , 5 CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* **********Create element for thin plate et,2,combin14 keyopt,2,3,2 ! UX, UY DOF r,2,1.7225e6 ! Spring constant k = EA/L; E= 344.5e6 Pa (A and L average measurements) !!!!mesh thin plate on PZT 1 TYPE, 2 REAL, 2 FLST,5,2,4,ORDE,2 FITEM,5,37 FITEM,5,114 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y
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LESIZE,_Y1, , ,1, , , , ,1 FLST,5,2,4,ORDE,2 FITEM,5,180 FITEM,5,199 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,1, , , , ,1 !!!mesh thin on PZT 1 MSHKEY,0 CM,_Y,AREA ASEL, , , , 3 CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !!!!mesh thin plate on PZT 2 FLST,5,2,4,ORDE,2 FITEM,5,171 FITEM,5,213 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,1, , , , ,1 FLST,5,2,4,ORDE,2 FITEM,5,221 FITEM,5,-222 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,1, , , , ,1 MSHKEY,0 CM,_Y,AREA ASEL, , , , 11 CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1
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CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !!!!mesh thin plate on PZT 3 FLST,5,2,4,ORDE,2 FITEM,5,104 FITEM,5,201 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,1, , , , ,1 FLST,5,2,4,ORDE,2 FITEM,5,219 FITEM,5,-220 CM,_Y,LINE LSEL, , , ,P51X CM,_Y1,LINE CMSEL,,_Y LESIZE,_Y1, , ,1, , , , ,1 MSHKEY,0 CM,_Y,AREA ASEL, , , , 10 CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* **********create element for MESHING COMPLIANT PIECE et,3,plane82 ! choose element type for compliant mechanism and end effector keyopt,3,3,2 ! PLANE STRAIN assumption mp,ex,3,117e9 ! modulus young (Pa) !compliant ! mp,nuxy,3,0.3 ! poisson ratio !properties! !* **********MESHING COMPLIANT PIECE TYPE,3 MAT,3 MSHAPE,1,2D
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FLST,5,3,5,ORDE,2 FITEM,5,12 FITEM,5,-14 CM,_Y,AREA ASEL, , , ,P51X CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 !* ***************Creating bolts' element***** ***************** et,4,plane2 ! choose element type for bolts keyopt,4,3,2 ! PLANE STRAIN ASSUMPTION mp,ex,4,358.28e6 ! modulus young (Pa) mp,nuxy,4,0.3 ! poisson ratio TYPE,4 MAT,4 !!!!!create nodes on KP bolts FLST,3,3,3,ORDE,3 FITEM,3,1 FITEM,3,49 FITEM,3,74 NKPT,0,P51X e,8290,448,450 e,8290,450,452 e,8290,452,441 e,8290,441,444 e,8290,444,446 e,8290,446,442 e,8290,442,461 e,8290,461,463 e,8290,463,454 e,8290,454,456 e,8290,456,458 e,8290,458,448 e,8291,5931,5933 e,8291,5933,5922 e,8291,5922,5925
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e,8291,5925,5927 e,8291,5927,5923 e,8291,5923,5942 e,8291,5942,5944 e,8291,5944,5935 e,8291,5935,5937 e,8291,5937,5939 e,8291,5939,5929 e,8291,5929,5931 e,8289,3192,3211 e,8289,3211,3213 e,8289,3213,3204 e,8289,3204,3206 e,8289,3206,3208 e,8289,3208,3198 e,8289,3198,3200 e,8289,3200,3202 e,8289,3202,3191 e,8289,3191,3194 e,8289,3194,3196 e,8289,3196,3192 !!!* **create elements for end-effector plate et,5,plane2 keyopt,5,3,2 mp,ex,5,69000000000 ! modulus young (Pa) !end effector ! mp,dens,5,7860 ! Density (kg/m^3) !properties ! mp,nuxy,5,0.3 ! poisson ratio type,5 mat,5 !* ****** create big end-effector circle CYL4, , ,35.1e-3 !* ****** create one small circle CYL4,6e-3,27e-3,2.5e-3 !! copy small circle CSYS,1 FLST,3,1,5,ORDE,1 FITEM,3,4
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AGEN,3,P51X, , , ,120, , ,0 !!! subtract all small circles FLST,3,3,5,ORDE,3 FITEM,3,4 FITEM,3,6 FITEM,3,-7 ASBA, 2,P51X /REPLOT CSYS,0 !!BACK TO CARTESIAN !!!create small circle for end-effector CYL4, , ,2.5e-3 !!!!substract small circle from end-effector plate ASBA,9, 2 !!mesh end-effector platform MSHKEY,0 CM,_Y,AREA ASEL, , , ,4 CM,_Y1,AREA CHKMSH,'AREA' CMSEL,S,_Y AMESH,_Y1 CMDELE,_Y CMDELE,_Y1 CMDELE,_Y2 MSHKEY,0 !* ****** create node for end-effector and circle N,9310,0,0 e,9310,8388,8399 e,9310,8399,8397 e,9310,8397,8395 e,9310,8395,8405 e,9310,8405,8403 e,9310,8405,8403 e,9310,8403,8401 e,9310,8401,8410 e,9310,8410,8408 e,9310,8408,8389 e,9310,8389,8393 e,9310,8393,8391
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e,9310,8391,8388 !!!!couple nodes of end-effector-bolts !!Bolt 1 cp,1,all,441,8316 cp,4,all,452,8327 cp,7,all,450,8325 cp,10,all,448,8323 cp,13,all,458,8333 cp,16,all,456,8331 cp,19,all,454,8329 cp,22,all,463,8338 cp,25,all,461,8336 cp,28,all,442,8317 cp,31,all,446,8321 cp,34,all,444,8319 !!Bolt 2 cp,37,all,5942,8384 cp,40,all,5923,8365 cp,43,all,5927,8369 cp,46,all,5925,8367 cp,49,all,5922,8364 cp,52,all,5933,8375 cp,55,all,5931,8373 cp,58,all,5929,8371 cp,62,all,5939,8381 cp,65,all,5937,8379 cp,68,all,5935,8377 cp,71,all,5944,8386 !!Bolt 3 cp,74,all,3208,8357 cp,77,all,3206,8355 cp,80,all,3204,8353 cp,83,all,3213,8362 cp,86,all,3211,8360 cp,89,all,3192,8341 cp,92,all,3194,8343 cp,95,all,3196,8345 cp,98,all,3191,8340 cp,102,all,3202,8351 cp,105,all,3200,8349
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cp,108,all,3198,8347 !!!FIXING BOLTS FINISH /SOL FLST,2,12,4,ORDE,6 FITEM,2,48 FITEM,2,-51 FITEM,2,115 FITEM,2,-118 FITEM,2,182 FITEM,2,-185 DL,P51X, ,ALL, sbctran !!! Shifting Element Coordinate System for PZTs 2 and 3 LOCAL,11,0,0,0,0,240, , ,1,1, !creation LOCAL,12,0,0,0,0,120, , ,1,1, !of new cs asel,s,,,8 !select area to be modified allsel,below,area !activate selection for area FINISH /PREP7 EMODIF,all,ESYS,11, !modify elements asel,s,,,5 allsel,below,area EMODIF,all,ESYS,12, allsel !select all instead of the previously chosen ones !!!!apply voltages on PZT1 /SOL FLST,2,1,4,ORDE,1 FITEM,2,136 DL,P51X, ,VOLT,PZT1 FLST,2,1,4,ORDE,1 FITEM,2,114 DL,P51X, ,VOLT, sbctran !!!apply voltages on pzt2 FLST,2,1,4,ORDE,1 FITEM,2,215 DL,P51X, ,VOLT, PZT2
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FLST,2,1,4,ORDE,1 FITEM,2,213 DL,P51X, ,VOLT, sbctran !!!apply voltages on PZT3 FLST,2,1,4,ORDE,1 FITEM,2,208 DL,P51X, ,VOLT,PZT3 FLST,2,1,4,ORDE,1 FITEM,2,201 DL,P51X, ,VOLT, sbctran P=4 !!!apply preloadforce on PZT1 FLST,2,2,3,ORDE,2 FITEM,2,210 FITEM,2,-211 FK,P51X,FX,-3.981eP sbctran !!!!apply preload on PZT2 FLST,2,2,3,ORDE,2 FITEM,2,217 FITEM,2,-218 FK,P51X,FX,0.5*2.56667eP FLST,2,2,3,ORDE,2 FITEM,2,217 FITEM,2,-218 FK,P51X,FY,0.866*2.566667eP sbctran !!!apply preload on PZT3 FLST,2,2,3,ORDE,2 FITEM,2,205 FITEM,2,212 FK,P51X,FX,0.5*5.3950eP FLST,2,2,3,ORDE,2 FITEM,2,205 FITEM,2,212 FK,P51X,FY,-0.866*5.3950eP sbctran finish
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/prep7 !!!!To obtain rotation !delete element in the center edele,4331,4343,1 et,6,beam3 KEYOPT,6,6,1 KEYOPT,6,9,0 KEYOPT,6,10,0 mp,ex,6,69e9 mp,dens,6,7860 mp,nuxy,6,0.3 r,6,0.01,8.33e-8,0.01 type,6 mat,6 real,6 !* ****** create elements for end-effector in the centre e,9310,8388 e,9310,8389 e,9310,8390 e,9310,8391 e,9310,8392 e,9310,8393 e,9310,8394 e,9310,8395 e,9310,8396 e,9310,8397 e,9310,8398 e,9310,8399 e,9310,8400 e,9310,8401 e,9310,8402 e,9310,8403 e,9310,8404 e,9310,8405 e,9310,8406 e,9310,8407 e,9310,8408 e,9310,8409 e,9310,8410
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e,9310,8411 /solu solve !preparation of new coordinate systems imitating experiment axis (comparison purpose) !/prep7 !csys,0 ! !dsys,0 !k,250,-10.2e-3,9.35e-3 !k,251,9.35e-3, 10.2e-3 !k,252,0,0,0 !KWPLAN,-1,252,250,251 !CSWPLA,1000,0,1,1, !csys,1000 !dsys,1000 !FINISH !/SOLU !PSTRES,on !SOLVE !FINISH !/POST1 !rsys,0 !NSEL,S,LOC,X,0,0,0 !prnsol,dof !csys,0 !dsys,0 !rsys,0 !finish !Modal analysis !/SOLU !ANTYPE,MODAL !MODOPT,LANB,5,0,0 ! BLOCK LANCZOS, EXTRACT 5 MODES !MXPAND,5 !PSTRES,on !solve
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E.5. The Modified MATLAB Codes
Rs = 29.546; % Length of OCi n=11; % Length of Lbc f1=1.1733; f2=0.81569; lab=17.720; fa=f2+asin( n*sin(f2)/lab ) fai=f1+f2; co = lab*sin(fa); R0=3.5; tt = [cos(fai+2*pi/3), sin(fai+2*pi/3), sin(fai)*Rs*1e3; cos(fai), sin(fai), sin(fai)*Rs*1e3; cos(fai+4*pi/3), sin(fai+4*pi/3), sin(fai)*Rs*1e3]; % unit is in um jme = -(co/R0) .* inv(tt) %jacobian matrix JL angle = 53.961*pi/180; coordinate_transform = [ cos(angle), -sin(angle), 0; sin(angle), cos(angle), 0; 0, 0, 1 ] ; jlca = [ 1.610797127, -1.298942876,-0.164324634; -0.6056938677, -1.42884239, 1.37702048; -3.269206385e-5, -2.830203574e-5 ,-2.019743117e-5 ] % matrix obtained by experiment JL = inv(coordinate_transform) * jme V1= 0; % D = 635 e-12 m/V * 130 * 1.4 *V (The manufacturer eqn) V2= 0 ; % V3= 32; % L1=(1.1557e-4)*V1; L2=(1.1557e-4)*V2; L3=(1.1557e-4)*V3; in = [ L1; L2; L3]; % input displacement of PZT 1, 2, and 3
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outme_transform = JL*in % results in x, y and r direction, unit in um, um, and rad outjca=jlca*in % experimental results %************************* calibrated constant result op_Rs=31.2969; op_lab=16.3333; op_n=10.7105; op_f1=1.0951; op_f2=0.3366; op_fai=op_f1+op_f2; op_fa=op_f2+asin( op_n*sin(op_f2)/op_lab ); op_co = op_lab*sin(op_fa); R0=3.5; op_tt = [cos(op_fai+2*pi/3), sin(op_fai+2*pi/3), sin(op_fai)*op_Rs*1e3; cos(op_fai), sin(op_fai), sin(op_fai)*op_Rs*1e3; cos(op_fai+4*pi/3), sin(op_fai+4*pi/3), sin(op_fai)*op_Rs*1e3]; op_jme = -(op_co/R0) .* inv(op_tt) angle = 53.961*pi/180; coordinate_transform = [ cos(angle), -sin(angle), 0; sin(angle), cos(angle), 0; 0, 0, 1 ] ; jlca = [ 1.610797127, -1.298942876,-0.164324634; -0.6056938677, -1.42884239, 1.37702048; -3.269206385e-5, -2.830203574e-5 ,-2.019743117e-5 ] ; op_JL = inv(coordinate_transform) * op_jme in = [ L1; L2; L3];
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opoutme_transform = op_JL * in % calibrated results in x,y and r direction,unit in um, um, and rad Jacobian = op_JL % jacobian matrix Jacob_transpose = transpose(Jacobian) ansys11= -15046404312.184;%input from ANSYS stiffness values ansys12= 15274759135.731 ; ansys13= -233931320.70317; ansys21= -14958247330.251; ansys22= 15182139498.29 ; ansys23= -229361866.55399 ; ansys31= -13227266.1 ; ansys32= 13155063.491757 ; ansys33= 67399.571490299 ; ansys_stiffness = [ansys11 ansys12 ansys13; ansys21 ansys22 ansys23; ansys31 ansys32 ansys33] total_stiffness = ansys_stiffness * Jacob_transpose* Jacobian ; system_stiffness = eig(total_stiffness) E.6. Four-Bar Mechanism Model
! The obtained frequency results in ANSYS are in Hertz. ! 1 Hertz = 6.2831853 radian/second /prep7 /title, case study of the four-bar mechanism (model 2) !!!Kinematics model of the four-bar mechanism phi=0 ! Input angle A2=-phi/57.29578 L1=10 L2=4.25 L3=11 L4=10.65 C2=COS(A2) S2=SIN(A2) XA=L2*C2 YA=L2*S2 L1XA=L1-Xa LX2=L1Xa*L1Xa YA2=YA*YA D2=LX2+YA2
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D=D2**0.5 CA=L1XA/D SA=-YA/D L32=L3**2 L42=L4**2 DL3=D*L3 DL32=DL3*2 D3=D2+L32 D34=D3-L42 CB=D34/DL32 CB2=CB*CB CB21=-CB2+1 SB=CB21**0.5 L3CA=L3*CA L3SA=L3*SA L3CC=L3CA*CB L3CS=L3CA*SB L3SS=L3SA*SB CCSS=L3CC-L3SS L3SC=L3SA*CB SCCS=L3SC+L3CS XB=XA+CCSS YB=YA+SCCS !!!Develop elements for links and joints ET,1,3 !mass element ET,2,21,,,4 !mass element with activation EX,1,10.3e6 DENS,1,.000254 R,1,.167,.0003881,.167 ! crank R,2,.063,.00002084,.063 ! coupler R,3,.000239 k,1,0,0 k,2,xa,ya k,3,xb,yb k,4,l1,0 KFILL,1,2,2,5,1,1, KFILL,2,3,2,7,1,1 KFILL,3,4,2,9,1,1 nkpt,1,1 nkpt,2,5 nkpt,3,6 nkpt,4,2
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nkpt,5,2 nkpt,6,7 nkpt,7,8 nkpt,8,3 nkpt,9,3 nkpt,10,9 nkpt,11,10 nkpt,12,4 e,1,2 e,2,3 e,3,4 real,2 e,5,6 e,6,7 e,7,8 e,9,10 e,10,11 e,11,12 type,2 real,3 e,5 e,9 cp,1,ux,4,5 cp,2,uy,4,5 cp,3,ux,8,9 cp,4,uy,8,9 d,1,ux,,,,,uy,rotz d,12,ux,,,,,uy !solving by static analysis /solu pstres,on solve finish !solving by use of modal analysis /solu antype,modal modopt, lanb, 3, mxpand,3 pstres,on
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solve ! Model 3 /prep7 /title, case study of the four-bar mechanism (model 3) PHI=0 ! Adjust angle A2=-phi/57.29578 L1=10 L2=4.25 L3=11 L4=10.65 C2=COS(A2) S2=SIN(A2) XA=L2*C2 YA=L2*S2 L1XA=L1-Xa LX2=L1Xa*L1Xa YA2=YA*YA D2=LX2+YA2 D=D2**0.5 CA=L1XA/D SA=-YA/D L32=L3**2 L42=L4**2 DL3=D*L3 DL32=DL3*2 D3=D2+L32 D34=D3-L42 CB=D34/DL32 CB2=CB*CB CB21=-CB2+1 SB=CB21**0.5 L3CA=L3*CA L3SA=L3*SA L3CC=L3CA*CB L3CS=L3CA*SB L3SS=L3SA*SB CCSS=L3CC-L3SS L3SC=L3SA*CB SCCS=L3SC+L3CS XB=XA+CCSS YB=YA+SCCS ET,1,3 !mass element ET,2,21,,,4 !mass element with activation EX,1,10.3e6
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DENS,1,.000254 R,1,.167,.0003881,.167 ! crank R,2,.063,.00002084,.063 ! coupler R,3,.000239 k,1,0,0 k,2,xa,ya k,3,xb,yb k,4,l1,0 KFILL,1,2,3,5,1,1, KFILL,2,3,3,8,1,1, KFILL,3,4,3,11,1,1, nkpt,1,1 nkpt,2,5 nkpt,3,6 nkpt,4,7 nkpt,5,2 nkpt,6,2 !will be for mass element nkpt,7,8 nkpt,8,9 nkpt,9,10 nkpt,10,3!will be for mass element nkpt,11,3 nkpt,12,11 nkpt,13,12 nkpt,14,13 nkpt,15,4 e,1,2 e,2,3 e,3,4 e,4,5 real,2 e,6,7 e,7,8 e,8,9 e,9,10 e,11,12 e,12,13 e,13,14 e,14,15
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type,2 real,3 e,6 e,10 cp,1,ux,5,6 cp,2,uy,5,6 cp,3,ux,10,11 cp,4,uy,10,11 d,1,ux,,,,,uy,rotz d,15,ux,,,,,uy !solving by static analysis /solu pstres,on solve finish !solving by use of modal analysis /solu antype,modal modopt, lanb, 3, mxpand,3 pstres,on solve E.7. Two-Bar Mechanism Model
!Define parameters x=10e-2 y=10e-2 /prep7 k,1,0,0 k,2,X,Y k,3,10e-2,0 nkpt,1,1 nkpt,2,2 nkpt,3,2 nkpt,4,3
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et,1,combin14 KEYOPT,1,3,2 r,1,1000 !10 N/cm = 10*100 type,1 mat,1 real,1 e,1,2 e,3,4 cp,1,ux,2,3 cp,2,uy,2,3 !Constrain node 1 /SOL d,1,ux,0 d,1,uy,0 !Constrain node 4 d,4,ux,0 d,4,uy,0 !Move CS to end-effector (P) NWPAVE,2 CSWPLA,11,0,1,1, csys,11 dsys,11 !Apply Fx=1 at node 2 or node 3 F,2,FX,1 solve finish /post1 rsys,11 *GET,DISPX_FX,NODE,2,u,x *GET,DISPY_FX,NODE,2,u,y finish /solu FDELE,2,FX !Apply Fy=1 at node 2 or node 3 F,2,FY,1
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solve finish /post1 rsys,11 *GET,DISPX_FY,NODE,2,u,x *GET,DISPY_FY,NODE,2,u,y /solu FDELE,2,FY *DIM, COMPLI, ARRAY,2,2 COMPLI(1,1)=DISPX_FX,DISPY_FX COMPLI(1,2)=DISPX_FY,DISPY_FY *DIM,STIFF,ARRAY,2,2 *MOPER,STIFF,COMPLI,INVERT !perform inversion !Present the results in the output window !Define parameters *DIM,LABEL,CHAR,1 LABEL(1) = '' ! LABEL(1) is unchangeable *DIM,VALUE,,2,2 ! *VFILL,VALUE(1,1),DATA,STIFF(1,1) *VFILL,VALUE(1,2),DATA,STIFF(1,2) *VFILL,VALUE(2,1),DATA,STIFF(2,1) *VFILL,VALUE(2,2),DATA,STIFF(2,2) /OUT,systiff,vrt !save values in 'systiff' parameter /COM,----------------------SYSTEM STIFFNESS MATRIX (2 x 2)----------------------------- *VWRITE,LABEL(1),VALUE(1,1),VALUE(1,2) (1X,A8,' ', F15.4, ' ', F15.4) *VWRITE,LABEL(1),VALUE(2,1),VALUE(2,2) (1X,A8,' ', F15.4, ' ', F15.4) /COM,---------------------------------------------------------------------------------- /OUT FINISH *LIST,systiff,vrt !produce values in 'systiff' parameter
